1 Introduction

We consider the degenerate parabolic system

$$\begin{array}{@{}rcl@{}} \partial_{t}u - {\Delta} u + \partial_{t}v &=& f\quad\text{ in}~ Q_{T}, \quad u\vert_{\partial{\Omega}} = 0,\quad u\vert_{t=0} = u^{0},\\ \partial_{t}v + \alpha v &=& \phi(u) \quad\text{ in}~ Q_{T},\quad v\vert_{t=0} = v^{0}, \end{array} $$
(1)

where Q T =Ω×(0,T] and Ω a bounded domain in \(\mathbb R^{2}\). The difficulty in this problem is twofold. First, typically the nonlinearity ϕ(⋅) is only Hölder continuous, with exponent 0<p<1, but nondecreasing. Second, in the case α>0, the system is degenerate as the second equation defining the variable v is only of first order. Problems of this type naturally arise in modeling flows through porous media either with slow (i.e., non-equilibrium) adsorption, α κ>0 and \(\phi (\cdot ) = \kappa \tilde {\phi }(\cdot )\) (see Knabner [10] and van Duijn and Knabner [7, 8]), or with irreversible chemical reaction in which case α=0 (see the literature cited in Nochetto [13]).

In the case α=0, the system (1) reduces to a scalar problem

$$\partial_{t}u - {\Delta} u + \phi(u) = f\quad\text{ in}~\; Q_{T},\quad u\vert_{\partial{\Omega}} = 0,\quad u\vert_{t=0} = u^{0}. $$

Our analysis covers this special case, too, where some of the technical difficulties arising from the degeneracy in the general problem do not occur. In particular, all results hold uniformly with respect to the final time T.

The other limit case where \(\phi (\cdot ) = \kappa \tilde {\phi }(\cdot )\), and α = κ→∞, leads us to the diffusion problem

$$ \partial_{t}\left( u+\tilde{\phi}(u)\right) - {\Delta} u = f \quad\text{ in}\;~ Q_{T}, \quad u\vert_{\partial{\Omega}} = 0,\quad u\vert_{t=0} = u^{0}. $$
(2)

This kind of problem arises in the modeling of flow through porous media with fast (i.e., equilibrium) adsorption (see Knabner [10] and van Duijn and Knabner [8]) and also may be looked at as a generalization of the “porous medium equation” after a change of variables (see the literature cited in Nochetto and Verdi [14]). Its discretization has been studied, for example, in Nochetto and Verdi [14] and Barrett and Knabner [1, 2]. In our analysis of the discretization of problem (1), we will trace the dependence of all constants on the parameter α and on a bound M for |ϕ(⋅)|. This will enable us to extract also some results for the discretization of the limit problem (2) from our results for problem (1); see Remark 7, below.

We note that problem (1) is to be viewed as a model case for more general and practically relevant situations and has been chosen only for simplifying the presentation. The results of this paper largely extend to the cases that

  • the Laplacian Δ is replaced by a general second-order uniformly elliptic operator A with coefficients depending on x and t,

  • the homogeneous Dirichlet boundary condition is replaced by a more general nonhomogeneous boundary condition of mixed type,

  • the nonlinear function ϕ(⋅) also depends on the variable v,

  • the time-derivative term has the form ∂ t (ρ u) with some positive coefficient ρ possibly depending on x and t.

We consider the spatial discretization of problem (1) by the (“continuous”) finite element Galerkin method leaving the time variable continuous. Here, approximations u h ≈u and v h ≈v are sought for as solutions of corresponding systems of ordinary differential equations in certain finite element subspaces \(V_{h} \subset V := \dot {W}^{1}_{2}({\Omega })\) and \(W_{h}\subset W:= {W^{1}_{2}}({\Omega })\), respectively. It turns out that these systems possess unique solutions for all time t≥0.

Most of the existing literature for degenerate parabolic equations deals with energy norm error estimates. We refer to Barrett and Knabner [1], for problem (1) with α = κ>0, \(\phi (\cdot ) = \kappa \tilde {\phi } (\cdot )\), and to Barrett and Knabner [2], and Nochetto and Verdi [14] for problem (2). The approach in Barrett and Knabner [1, 2] is based on “energy techniques” developed in Knabner [10] (see also van Duijn and Knabner [8]), combined with a regularization of ϕ(⋅). The available L 2-error estimate

$$ \|u-u_{h}\|_{2;Q_{T}} \le ch^{1+p}, $$
(3)

with p being the Hölder exponent of ϕ(⋅), is only suboptimal. In case of non-degeneracy conditions (see Remark 8, below), this order can be improved, but still remains suboptimal. On the other hand the optimal-order error estimate

$$ \|\nabla(u-u_{h})\|_{2;Q_{T}} \le ch $$
(4)

is available. It was suspected that by other than L 2 techniques, the estimate (3) could be improved to the optimal order \(\mathcal {O}(h^{2})\). This is the general theme of the present paper.

The main result is the following L ∞ -error estimate of almost optimal order:

$$ \|u-u_{h}\|_{\infty;Q_{T}} \le c(T,\alpha,K,M)\, h^{2}|\ln(h)|^{2}. $$
(5)

The parameters M and K describe certain assumed bounds for the nonlinear function ϕ(⋅), and for the data of problem (1). There is no explicit dependency on the regularity of the solution {u,v} itself. The proof of estimate (5) is based on linearization through a parabolic duality argument introduced in Luskin and Rannacher [12]. In this case, the standard approach (see Chen [3] and Johnson et al. [9]) to the finite element approximation of semilinear problems cannot be used due to lacking Lipschitz continuity of the nonlinearity ϕ(⋅). To overcome this difficulty in the linearization, we use the difference quotient

$$b := \frac{\phi(u)-\phi(u_{h})}{u-u_{h}}, $$

rather than the (non-existing) derivative ϕ ′(⋅) as the coefficient in the “dual” problem. This trick was originally introduced in Nochetto [13] for treating the stationary analogue of the scalar problem (2). Since ϕ(⋅) is not Lipschitz continuous, the function b is not bounded but satisfies b≥0 due to the monotonicity of ϕ(⋅). The latter guarantees that the solutions of the auxiliary problems admit sharp a priori estimates in certain L 1 spaces. Via the duality between L 1 and L ∞ , this then leads us to the optimal-order pointwise error estimate (5). Here, the argument is based on sharp L ∞ -error estimates for the finite element approximation of regular linear parabolic problems which are available from Schatz et al. [17], Rannacher [15], and Chen [3].

2 The Model Problem

Let Ω be a bounded domain in \(\mathbb R^{2}\) with smooth boundary ∂Ω. For technical simplicity only, Ω is assumed to be convex. Furthermore, let [0,T] be a fixed time interval and let Q t :=Ω×(0,t] for t∈(0,T]. We consider the following initial-boundary value problem: Find a pair of functions {u(x,t),v(x,t)} satisfying

$$\begin{array}{@{}rcl@{}} \partial_{t}u - {\Delta} u + \partial_{t}v &=& f \quad \text{in}~ Q_{T},\quad u\vert_{\partial{\Omega}}=0,\quad u\vert_{t=0} = u^{0},\\ \partial_{t}v + \alpha v &= &\phi(u)\quad\text{in}~ Q_{T},\quad v\vert_{t=0} = v^{0}. \end{array} $$
(6)

We use the standard notation L p (Ω) and \({W^{m}_{p}}({\Omega })\) for the Lebesgue and Sobolev spaces over Ω, respectively, and ∥⋅∥ p =∥⋅∥ p;Ω, ∥⋅∥ m,p =∥⋅∥ m,p;Ω for the corresponding norms. \(\dot {W}^{1}_{p}({\Omega })\) is the closure of \(C_{0}^{\infty }({\Omega })\) in \({W^{1}_{p}}({\Omega })\). The L 2 scalar product is simply denoted by (⋅,⋅). Furthermore, L p (0,T;H) are spaces of time-dependent functions v with values in a function space H, such that ∥v(⋅)∥ H is integrable to the pth power over (0,T). The Lebesgue spaces over the space-time cylinder Q T are denoted by L p (Q T ) with the norms \(\|\cdot \|_{p;Q_{T}}\). \(W^{2,1}_{p}(Q_{T})\) is the subspace of L p (Q T ) of functions with derivatives up to first-order in t and second-order in x belonging to L p (Q T ). Finally, we set \(V := \dot {W}^{1}_{2}({\Omega })\) and \(W := {W^{1}_{2}}({\Omega })\).

We formulate the following assumptions about problem (6), which will be used in various combinations, below.

  1. (A1)

    ϕ(⋅) is continuous on \(\mathbb R\).

  2. (A2)

    ϕ(⋅) is monotone, i.e., \((\phi (\xi )-\phi (\eta ))(\xi -\eta ) \ge 0, \quad \xi ,\eta \in \mathbb R\).

  3. (A3)

    The data of problem (6) satisfy α≥0 constant and

    $$f\in L^{\infty}(Q_{T}),\quad u^{0}\in V\cap W^{2,\infty}({\Omega}), \quad v^{0}\in L^{\infty}({\Omega}). $$

In Theorem 1, below, we show that weak solutions exist in the typical case of a nonlinearity ϕ(⋅), which is only Hölder continuous. This class of problems contains cases in which the solution has finite speed of propagation (see Knabner [10], van Duijn and Knabner [7], and van Duijn and Knabner [8]), i.e., a solution, which vanishes in a subregion initially will preserve this property for some time interval. This induces the occurrence of a free interface defined by supp(u). In addition, to the conditions (A1)–(A3), we make the following assumptions:

  1. (A4)

    ϕ∈C 1((−∞,0)∪(0,∞)), ϕ(0)=0, ϕ(s)>0 for s>0.

  2. (A5)

    There exist L, ε 0>0, and p∈(0,1], such that

    $$|\phi(a)-\phi(b)| \le L|a-b|^{p},\quad a,b\in [0,\varepsilon_{0}]. $$

Theorem 1

Under the assumptions (A1)–(A5), problem ( 6 ) possesses a unique solution {u,v} in the usual weak sense on Q T satisfying

$$ u\in W^{2,1}_{q}(Q_{T}), q\in [1,\infty),\quad v\in C^{1}([0,T],L_{\infty}({\Omega})). $$
(7)

Furthermore, there exist bounds \(\underline {u}, \overline {u}, \underline {v},\overline {v}\) such that

$$ \underline{u} \le u \le \overline{u},\quad \underline{v}\le v \le \overline{v}, $$
(8)

and the bounds \(\underline {u}\) and \(\overline {u}\) depend only on \(\|f\|_{\infty ;Q_{T}}\) and ∥u 0 ∥ ∞ .

Proof

For the proof of existence and uniqueness, we refer to Barrett and Knabner [1], (see also Knabner [10] for a more general problem with flux boundary condition). There, the case α>0 is treated by showing existence for a regularized problem including the bounds (8) in Barrett and Knabner [1], the convergence of the regularization process (see Barrett and Knabner [1](@var1@Lemma 2.1@var1@)), and thus finally existence and uniqueness for problem (6) (see Barrett and Knabner [1](@var1@Theorem 2.2@var1@)). Inspection of the proofs shows that the same results are valid for α=0. The improved regularity (7) can be obtained by standard parabolic regularity theory (see Knabner [10], Part II, Theorems 3.2 and 3.5). □

3 Finite Element Galerkin Approximation

For the discretization parameter h∈(0,1), let \(\mathbb T_{h}\) be finite decompositions of Ω into triangles or (convex) quadrilaterals K, where \(h := \max \{h_{K},K\in \mathbb T_{h}\}\) for h K :=diam(K). Furthermore, let \({\Omega }_{h} = \cup \{K\in \mathbb T_{h}\}\). For the family \(\{\mathbb T_{h}\}_{h>0}\) of decompositions, we impose the usual regularity conditions (see Ciarlet [4]):

  1. (T1)

    Any two cells \(K,K^{\prime }\in \mathbb T_{h}\) either meet in vertices or in entire common sides and all vertices of Ω h lie on ∂Ω (“mesh conformity property”).

  2. (T2)

    There exists a constant c 2>0 such that \(\min \{h_{K}, K\in \mathbb T_{h}\} \ge c_{2}h\) (“uniform size property”).

  3. (T3)

    There exists a constant c 1>0 such that each cell \(K\in \mathbb T_{h}\) contains a circle with radius ρ K ≥c 1 h K (“uniform shape property”).

Remark 1

These strong assumptions on the regularity of the decompositions \(\{\mathbb T_{h}\}_{h>0}\) are made mainly for simplicity. In particular, all the results of this paper remain valid if the conditions (T2) and (T3) are relaxed to \(\rho _{K}\ge c_{1}h_{K}^{\gamma }\) and \(\min \{h_{K},K\in \mathbb T_{h}\} \ge c_{2} h^{\gamma }\), respectively, with some constant γ>1 (see Rannacher [16] for a discussion of this point). These relaxed conditions allow for local mesh refinement with polynomial rate of mesh size decrease.

Corresponding to the decompositions \(\mathbb T_{h}\), we introduce the finite element spaces

$$\begin{array}{@{}rcl@{}} V_{h} &:=& \{\phi_{h}\in \dot{W}_{2}^{1}({\Omega}_{h}), \phi_{h}\vert_{K}\in P(K), K\in\mathbb T_{h}\},\\ W_{h} &:=& \{\psi_{h}\in {W_{2}^{1}}({\Omega}_{h}), \psi_{h}\vert_{K}\in P(K), K\in\mathbb T_{h}\}, \end{array} $$

where P(K): = P 1(K) for triangular elements and P(K): = Q 1(K) for (isoparametric) quadrilateral elements. If necessary, the functions in V h and W h are understood to be extended to \(\overline {\Omega }\) in such a way that \(V_{h}\subset \dot {W}_{2}^{1}({\Omega })\) and \(W_{h}\subset {W_{2}^{1}}({\Omega })\cap L_{\infty }({\Omega })\). In particular, we require that V h ⊂W h . We denote by P h and Q h the L 2 projections onto V h and W h , respectively. Furthermore, a discrete analogue Δ h :V h →V H of the Laplacian is defined through the relation

$$-({\Delta}_{h} w_{h},\phi_{h}) = (\nabla w_{h},\nabla\phi_{h})\quad\forall\phi_{h}\in V_{h}. $$

With this notation, the approximate problem reads as follows: Find pairs {u h ,v h }∈C 1([0,T];V h )×C 1([0,T];W h )satisfying

$$\begin{array}{@{}rcl@{}} \partial_{t}u_{h} - {\Delta}_{h}u_{h} + \partial_{t}v_{h} &= P_{h}f\quad on~ (0,T],\quad u_{h}(0) = P_{h}u^{0},\\ \partial_{t}v_{h} + \alpha v_{h} &= Q_{h}\phi(u_{h})\quad on~ (0,T],\quad v_{h}(0) = Q_{h}v^{0}. \end{array} $$
(9)

Theorem 2

Under the assumptions (A1), (A2), and (T1)–(T3), the semidiscrete problems ( 9 ) possess unique solutions {u h ,v h }, which exist for all time t≥0.

Proof

We use the argument developed in Knabner [10] and Barrett and Knabner [1, 2]. We may assume that ϕ(0)=0, as we can achieve this property by the transformations \(\tilde {\phi }(u) := \phi (u)-\phi (0)\), \(\tilde {v} := v-\alpha ^{-1}\phi (0)\) for α>0, or \(\tilde {f} := f-\phi (0)\) for α=0, respectively. The equations (9) can be rewritten as a system of ordinary differential equations for the nodal values of u h and v h with continuous right-hand side. Therefore, existence locally in time can be assured by Peano’s theorem. To show global existence, it is sufficient to deduce L ∞ bounds for u h ,v h , which may depend on h. Let \(\{{u_{h}^{i}}, {v_{h}^{i}}\}\) be solutions of (9) for data f i,u 0,i,v 0,i,i=1,2, which will be considered on their common existence interval. Set \(u_{h} := {u_{h}^{1}}-{u_{h}^{2}}, v_{h} := v_{h}^{1}-{v_{h}^{2}}\), etc.

  1. (i)

    First, we consider the case α>0. We differentiate the first equation in (9), use the test function

    $$\phi_{h}(x,s) := {{\int}_{s}^{t}} u_{h}(x,r)dr, $$

    integrate over [0,t] and integrate by parts in space. This leads us to

    $$\begin{array}{@{}rcl@{}} {{\int}_{0}^{t}} \|u_{h}\|^{2}ds + {{\int}_{0}^{t}}(v_{h},u_{h})ds &+& \frac{1}{2}\left\|{{\int}_{0}^{t}} \nabla u_{h}ds\right\|^{2}\\ &=& {{\int}_{0}^{t}} (f,\phi_{h})ds + {{\int}_{0}^{t}} (u^{0}+v^{0},u_{h})ds. \end{array} $$

    Then, by integration by parts in time and separation on the right-hand side,

    $$\|u_{h}\|_{2;Q_{t}}^{2} + {{\int}_{0}^{t}}(v_{h},u_{h})ds \le c\left\{t^{2}\|f\|_{2;Q_{t}}^{2} + t\|u^{0}+v^{0}\|_{2;{Q_{t}^{2}}}\right\}. $$

    Here, the constant c is independent of the data of the problem. Next, we differentiate the second equation in (9), use the test function ϕ h : = u h , and integrate over [0,T]. Due to assumption (A2) and V h ⊂W h , we arrive at

    $$\begin{array}{@{}rcl@{}} {{\int}_{0}^{t}} (v_{h},u_{h})ds &=& \frac{1}{\alpha}\left\{{{\int}_{0}^{t}}(\phi({u_{h}^{1}})-\phi({u_{h}^{2}}),{u_{h}^{1}}-{u_{h}^{2}})ds - {{\int}_{0}^{t}}(\partial_{t}v_{h},u_{h})ds\right\} \\ &\ge& -\frac{1}{\alpha}{{\int}_{0}^{t}}(\partial_{t}v_{h},u_{h})ds. \end{array} $$

    Using the test function ϕ h : = u h in the first equation of (9), we obtain

    $$-{{\int}_{0}^{t}}(\partial_{t}v_{h},u_{h})ds = \frac{1}{2}\left\{\|u_{h}(t)\|^{2} - \|{u_{h}^{0}}\|^{2}\right\} + {{\int}_{0}^{t}}\|\nabla u_{h}\|^{2}ds - {{\int}_{0}^{t}}(f,u_{h})ds. $$

    Combining the foregoing results yields

    $$\begin{array}{@{}rcl@{}} &&\|u_{h}\|_{2;Q_{t}}^{2} + \alpha^{-1}\left\{\|u_{h}(t)\|_{2;{\Omega}}^{2}+ \|\nabla u_{h}(t)\|_{2;{\Omega}}^{2} \right\}\\ &&\qquad\le c\left\{(t^{2}+\alpha^{-2})\|f\|_{2;Q_{t}}^{2} + t\|u^{0}+v^{0}\|_{2;{\Omega}}^{2} + \alpha^{-1}\|u^{0}\|_{2;{\Omega}}^{2} \right\}. \end{array} $$
    (10)

    We now set the second data set to zero, i.e., \({u_{h}^{2}} = {v_{h}^{2}} = 0\). Then, the last inequality particularly implies the bound

    $$\sup_{[0,t]}\|u_{h}\|_{2;{\Omega}} \le c(\alpha,t), $$

    and thus,

    $$ \sup_{[0,t]}\|u_{h}\|_{\infty;{\Omega}} \le c(\alpha,t,h). $$
    (11)

    Taking the test function ϕ = u h in the second equation in (9) gives us

    $$\|v_{h}\|_{2;Q_{t}}^{2} + \alpha^{-1}\|v_{h}(t)\|_{2;{\Omega}}^{2} \le c\alpha^{-1}\|\phi(u_{h})\|_{2;Q_{t}}^{2}, $$

    i.e., by (11) and assumption (A1),

    $$ \sup_{[0,t]} \|v_{h}\|_{\infty;{\Omega}} \le c(\alpha,t,h). $$
    (12)

    Thus, for each interval [0,T], (11) and (12) yield the desired bounds, which imply global existence of the approximate solutions. Concerning uniqueness, consider two solutions for the same data set. Then, by (10), \({u_{h}^{1}}= {u_{h}^{2}}\), and by the second equation in (9), also \({v_{h}^{1}} = {v_{h}^{2}}\).

  2. (ii)

    For α=0, the reasoning is a simplification of the above one. Inequality (10) is substituted by

    $$\|u_{h}(t)\|_{2;{\Omega}}^{2} + \|\nabla u_{h}\|_{2;Q_{t}}^{2} \le c\left\{\|f\|_{2;Q_{t}}^{2}+\|u^{0}\|_{2;{\Omega}}^{2} \right\}, $$

    which follows from using the test function ϕ h = u h in the equivalent formulation

    $$\partial_{t}u_{h} - {\Delta}_{h}u_{h} + P_{h}\phi(u_{h}) = P_{h}f. $$

    This completes the proof.

□

In our error analysis for the approximation (9), we assume that the nonlinearity ϕ(⋅) is uniformly bounded,

$$M := \sup_{\xi\in\mathbb R}|\phi(\xi)| < \infty. $$

This assumption is made only for technical convenience and can be dismissed by a standard argument; see Remark 4, below. We further introduce the quantity

$$K := \max\left\{\|u^{0}\|_{2,\infty}, \|v^{0}\|_{\infty}, {\sup}_{(0,T]}\|f\|_{\infty} \right\}. $$

Theorem 3

Suppose that the conditions (A1)–(A3), (A6), and (T1)–(T3) are satisfied. Then, for the solutions of the approximate problems ( 9 ) there holds

$$ \|u-u_{h}\|_{\infty;Q_{T}} \le Ch^{2}|\log(h)|^{2}, $$
(13)

where C=c(1+K+M)(1+αT), and c is a numerical constant depending only on Ω and on the characteristics of the decompositions \(\mathbb T_{h}\).

The lengthy proof of this theorem will be given in the next section. We note that the error estimate (13) does not contain any explicit bound on the regularity of the solution {u,v}. These are implicitly contained in the assumptions on Ω and on the data of problem (1). Furthermore, besides the bound (A8) and the monotonicity, there is no additional assumption on the nonlinear function ϕ(⋅); in particular, the Hölder exponent of ϕ does not occur. The order of convergence in (13) is confirmed by the numerical tests reported in Section 5, below.

Remark 2 (Estimate of v−v h )

In order to estimate the error v−v h , we need to assume a certain degree of regularity for v, say \(v\in L_{\infty }(0,T;C^{0,2p}(\overline {\Omega }))\), where p∈(0,1) is the assumed Hölder exponent of ϕ(⋅). Through the coupling between the unknowns u and v in the second equations of (1) and (9), the estimate (13) directly implies the corresponding one for v−v h . Starting with the error identity

$$\partial_{\tau}(Q_{h}v-v_{h}) + \alpha(Q_{h}v-v_{h}) = Q_{h}\phi(u) - Q_{h}\phi(u_{h}), $$

we conclude that

$$\|Q_{h}v-v_{h}\|_{\infty;Q_{T}} \le c\min\{T,\alpha^{-1}\} \|Q_{h}(\phi(u)-\phi(u_{h}))\|_{\infty;Q_{T}}. $$

Consequently, by the boundedness of the L 2-projection Q h with respect to the L ∞ -norm (see Douglas et al. [6]),

$$\|v-v_{h}\|_{\infty;Q_{T}} \le \|v-Q_{h}v\|_{\infty;Q_{T}} + c\min\{T,\alpha^{-1}\} \|\phi(u)-\phi(u_{h})\|_{\infty;Q_{T}}. $$

Then, for the assumed regularity of v and in virtue of the estimate (3) for u−u h , we obtain the following result for v−v h ;

$$\|v-v_{h}\|_{\infty;Q_{T}} \le Ch^{2p}|\log(h)|^{2p}. $$

The numerical tests presented in Section 5, below, indicate that these orders of convergence are best possible in the general situation. For special nonlinearities such as ϕ(s)=sgn(s)|s|p, which are Lipschitzian for positive arguments, and positive u and u h , the estimate (4) becomes

$$\|v-v_{h}\|_{\infty;Q_{T}} \le \|v-Q_{h}v\|_{\infty;Q_{T}} + Ch^{2}|\log(h)|^{2}. $$

Hence, in this case, the resulting order of convergence essentially depends on the regularity of v.

Remark 3 (Behavior for T→∞)

If f∈L ∞ (0,∞;L ∞ (Ω)), the constants in the error estimates (3) and (4) grow linearly with T. In the case α=0, they remain bounded as T→∞. Whether this may also be true in the general case α>0 is an open question.

Remark 4 (Unbounded ϕ(⋅))

The assumption that the nonlinear function ϕ(⋅) is uniformly bounded has been made only for technical convenience in order to separate the complications caused by the possible unboundedness of ϕ(ξ), as ξ→∞, from the more interesting problem of how to deal with the irregularity at ξ=0. In the general case of an unbounded ϕ(⋅), there occurs the problem of controlling the quantity \(M_{T,h} := \|\phi (u_{h})\|_{Q_{T}}\). In fact, as we will see below, what is actually needed in the proof of Theorem 3 is a bound for M T,h , which is obtained there simply by using the assumed global bound M for ϕ(⋅). To get rid of this assumption, one may use a standard continuation argument on the expense of introducing a restriction on the mesh size h depending on T.

Remark 5 (Lower order finite elements)

The use of C 0-elements for the variable v appears somewhat unnatural. In fact, in practical computations, one may prefer to work with discontinuous trial functions for v, which (similar to “mass lumping”) decouples the second equation in (6) for the corresponding nodal values. Furthermore, the evaluation of the integrals involving ϕ(u h ) usually requires numerical integration. As a simple example of this situation, we will consider a modification of the approximation (9), which employs the trial spaces

$$\tilde{W}_{h} = \{v_{h}\in L_{\infty}({\Omega}_{h}), v_{h}\vert_{K}, K\in \mathbb T_{h}\}. $$

In this case, \(V_{h}\not \subset \tilde {W}_{h}\). Due to the weak regularity of the function ϕ(⋅), the error introduced through this modification causes a significant reduction in the asymptotic accuracy of the approximation. Without taking quadrature into account, we conjecture that the following error estimate holds true:

$$\|u-\tilde{u}_{h}\|_{\infty;Q_{T}} \le \tilde{C}(T,\alpha,K,M) h^{1+p}, $$

where p∈(0,1) is the Hölder exponent of the function ϕ(⋅).

Remark 6 (Time discretization)

For discretizing the semidiscrete problem (9) with respect to time, one may use the backward Euler method. Denoting the time step size by k, the fully discrete approximations \(\{{U_{h}^{n}},{V_{h}^{n}}\}\in V_{h}\times W_{h}\) are determined by the equations

$$\begin{array}{@{}rcl@{}} k^{-1}({U_{h}^{n}}-U_{h}^{n-1}) - {\Delta}_{h}{U_{h}^{n}} + k^{-1}P_{h}({V_{h}^{n}}-V_{h}^{n-1}) &=&P_{h}f^{n},\quad n\ge 1,\\ k^{-1}({V_{h}^{n}}-V_{h}^{n-1}) + \alpha {V_{h}^{n}} &=& Q_{h}\phi({U_{h}^{n}}), \quad n\ge 1 \end{array} $$

with the initial values \({U_{h}^{0}} = P_{h}u^{0}\) and \({V_{h}^{0}} = Q_{h}v^{0}\). We think that the arguments for proving Theorem 3 carry over to this time stepping scheme leading, under our assumptions, to an error estimate of the form

$$\max_{0\le t_{n}\le T} \|u(t_{n})-{U_{h}^{n}}\|_{\infty} \le C\left\{h^{2}|\log(h)|^{2} + k|\log(k)|^{2} \right\}. $$

For higher-order methods, such as the Crank–Nicolson scheme, the argument of proof becomes more involved and is a yet open problem.

Remark 7 (Limit κ→∞)

As mentioned in Section 1, for \(\phi = k\tilde {\phi }\), the limiting process α = κ→∞ leads to a nonlinear diffusion problem of the form

$$ \partial_{t}(u+\tilde{\phi}(u)) - {\Delta} u = f\quad\text{in}\;~Q_{T},\quad u\vert_{\partial{\Omega}}=0, \quad u\vert_{t=0}=u^{0}. $$
(14)

In order to apply our error analysis to the finite element approximation of this problem, we need to trace the dependency of the constant in the estimate (3) on the parameter κ. Clearly, setting \(\tilde {M} := \sup _{\xi \in \mathbb R}|\tilde {\phi }(\xi )|\), there holds \(M=\kappa \tilde {M}\), and for α = κ, the constant in (3) takes the form

$$C(k) = c(1+K+\kappa\tilde{M})(1+\kappa T) = \mathcal{O}(\kappa^{2}). $$

Therefore, denoting the solution of (1) by u κ, and that of (14) by u ∞, we have the estimate

$$ \|u^{\kappa}-u_{h}^{\kappa}\|_{\infty;Q_{T}} \le C(T,K)\kappa^{2}h^{2}|\log(h)|^{2}, $$
(15)

which is independent of any further bound on the regularity of u κ. Now, for large κ, the semidiscrete problems (9) may be viewed as the approximation of a regularization of the limit problem (14). The error estimate (15) then directly implies estimates or \(u-u_{h}^{\kappa }\), provided that a bound for the error u−u κ is valid. Unfortunately, in this respect, only rather weak results are available. It is known that (see Barrett and Knabner [2]),

$$ \|u-u^{\kappa}\|_{2;Q_{T}} = \mathcal{O}(\kappa^{-(1+p)/2}), $$
(16)

provided (A4) and (A5) are satisfied. This estimate, upon balancing the factors κ 2 h 2|log(h)|2 and κ −(1 + p)/2, then results in

$$\|u-u^{\kappa}\|_{2;Q_{T}} = \mathcal{O}((h|\log(h)|^{2(1+p)/(5+p)}) $$

for κ = c(h|log(h)|)−4/(5 + p). This estimate is weaker than the ones obtained in Barrett and Knabner [2], and Nochetto and Verdi [14]. In view of the available analytical results for special solutions (see van Duijn and Knabner [8]), one may suppose that at best

$$ \|u-u^{\kappa}\|_{\infty;Q_{T}} = \mathcal{O}(\kappa^{-1}). $$
(17)

Note that this is rather speculative and a rigorous proof is not even in sight, yet. Combining (15) and (17), we arrive at the still suboptimal estimate

$$\|u-u^{\kappa}\|_{\infty;Q_{T}} = \mathcal{O}((h|\log(h)|)^{2/3}) $$

for κ = c(h|log(h)|)−2/3. Improving on this result through improving the perturbation estimate (16) is an interesting problem for further analytical studies.

Remark 8 (Free boundary approximation)

If a non-degeneracy condition holds, then also an approximation for the continuous free boundary

$$F = \partial\left\{(x,t)\in Q_{T}: u(x,t) > 0\right\} \cap Q_{T} = \partial Q_{T}^{+}\cap Q_{T} $$

is available. The non-degeneracy condition describes the minimal growth of u away from the front. The local behavior of the profile has only been analyzed for traveling wave solutions (see van Duijn and Knabner [8]). This leads us to the following conjecture: If conditions (A5), (A6) hold and (A6) is sharp, i.e.,

$$\phi(u) \ge cu^{p},\quad u\in [0,\delta] $$

for some c,δ>0, then

$$\text{meas}(Q_{T}^{+}{\Delta} Q_{T,h}^{+}) \le C(h|\log(h)|)^{2\beta}, $$

where the approximate free boundary is defined by

$$F_{h} := \partial\left\{(x,t)\in Q_{T}:\; u_{h}(x,t) > \delta \right\}\cap Q_{T} = \partial Q_{T,h}^{+}\cap Q_{T}, $$

and δ: = C ∗(h|log(h)|)2, C ∗ bigger than the constant in (3).

4 Proof of the Main Theorem

As discussed in Section 3, we may assume without loss of generality that the function ϕ(⋅) is uniformly bounded,

$$M := \sup_{\xi\in\mathbb R} |\phi(\xi)| < \infty. $$

Since, according to Theorem 1, the approximate solution u h exists on the time interval [0,T], we can fix the quantity

$${\Gamma} = {\Gamma}(T,h) := \max_{[0,T]}\|u_{h}\|_{\infty} + \max_{[0,T]}\|u\|_{\infty}, $$

(compare (8)), and introduce the notation \(B_{r} := \{\xi \in \mathbb R, |\xi |\le r\}\). The function ϕ(⋅) is uniformly continuous on B Γ. Hence, there are numbers ρ = ρ(T,Γ)>0, such that

$$\xi, \eta \in B_{\Gamma},\quad|\xi-\eta| \le \rho \quad\Rightarrow\quad |\phi(\xi)-\phi(\eta)| \le \frac{h^{2}}{T}. $$

Following an idea from Nochetto [13] for the stationary case, we introduce the function

$$b := \frac{\phi(u)-\phi(u_{h})}{\text{sign}(u-u_{h})\max\{|u-u_{h}|,\rho\}} $$

on Q T , where sign(ξ)=1 for ξ≥0 and sign(ξ)=−1 for ξ<0. Clearly, b≥0, since ϕ(⋅) is nondecreasing, and the norm \(\|b\|_{\infty ;Q_{T}}\) exists but it depends on h and T. However, we note that all the estimates derived below will be independent of this norm. Furthermore, by construction there holds

$$ \|b(u-u_{h}) - \{\phi(u)-\phi(u_{h})\}\|_{\infty;Q_{T}} \le \frac{h^{2}}{T}. $$
(18)

Remark 9 (Regularization)

Our definition of the function b differs from that used in Nochetto [13] as it applies to the nonlinear function ϕ(⋅) itself. This gives us an error estimate for the original problem rather than for a regularized form involving a nonlinearity ϕ ε (⋅) being smoothed on an interval of length ε. Thus, we do not require any restriction on those points at which Lipschitz continuity is lacking. Furthermore, there is no need for an estimate of \(\|u-u_{\varepsilon }\|_{\infty ;Q_{T}}\) in terms of ε, where {u ε ,v ε } is the solution of the regularized problem. An estimate of this type is here much harder to achieve than for the problem treated in Nochetto [13]. The quantity ρ is needed in the definition of b in order to compensate for the missing Lipschitz continuity of ϕ(⋅).

Using the above notation, the error equations for e u : = u−u h and e v : = v−v h now read as follows:

$$\begin{array}{@{}rcl@{}} (\partial_{t}e_{u},\phi_{h}) + (\nabla e_{u},\nabla\phi_{h}) + (be_{u},\phi_{h}) - \alpha(e_{v},\phi_{h}) &=& -(e_{b},\phi_{h}),\quad \phi_{h}\in V_{h}, \\ (\partial_{t}e_{v},\chi_{h}) - (be_{u},\chi_{h}) + \alpha(e_{v},\chi_{h}) &=& (e_{b},\chi_{h}),\quad \chi_{h}\in W_{h}, \end{array} $$

where e b is such that

$$\|e_{b}\|_{\infty;Q_{T}} \le \frac{h^{2}}{T}. $$

Thus, below, the linearization of problem (1) about its solution {u,v} will be based on the formal adjoint of the matrix operator

$$B = \left( \begin{array}{cc} -{\Delta} + a &\quad -\alpha \\ -b &\quad \alpha \end{array}\right). $$

Now, let any time t∈(0,T] and any function \(\chi \in C_{0}^{\infty }({\Omega })\) with ∥χ∥1≤1 be arbitrarily chosen but fixed. We intend to prove the estimate

$$ |((u-u_{h})(t),\chi)| \le c(1+\max\{K+M,\alpha K\})(1+\alpha t) h^{2}|\log(h)|^{2}, $$
(19)

uniformly with respect to χ. Taking then the suppremum over \(\chi \in C_{0}^{\infty }({\Omega })\), we obtain the desired error estimate (3),

Next, we prepare for the linearization of the given problem about its solution {u,v} through a parabolic duality argument; see Luskin and Rannacher [12] and Rannacher [15] for a similar approach in the context of linear parabolic problems. We define a pair of functions {G,H}={G (t),H (t)} as the solution of the following “dual” problem, running backward from time τ = t to τ=0,

$$\begin{array}{@{}rcl@{}} -\partial_{\tau} G - {\Delta} G &=& -b(G-H)\quad\text{in}~ Q_{t},\quad G\vert_{\partial{\Omega}}=0,\quad G\vert_{\tau=t}=\chi, \\ -\partial_{\tau} H &=& \alpha(G-H)\quad\text{in}~ Q_{t},\quad H\vert_{\tau=t}=0. \end{array} $$
(20)

Clearly, this problem is well-posed and possesses a unique classical solution. The following lemma provides the key a priori estimate for this Green-type function.

Lemma 1

For the solution {G,H} of problem ( 20 ), there holds the bound

$$ {{\int}_{0}^{t}} \|b(G-H)\|_{1}d\tau + \frac{1}{t}{{\int}_{0}^{t}} \|G-H\|_{1}d\tau\le 2(1+\alpha t). $$
(21)

Proof

We will use the regularized sign function

$$\text{sign}_{\varepsilon}(v) := \frac{v}{\sigma_{\varepsilon}(v)},\quad \varepsilon >0, $$

where σ ε (v):=(v 2 + ε 2)1/2. There holds

$$ \partial_{\tau}\sigma_{\varepsilon}(v) = \text{sign}_{\varepsilon}(v)\partial_{\tau} v,\quad \nabla\text{sign}_{\varepsilon}(v) = \frac{\varepsilon^{2}}{\text{sign}_{\varepsilon}(v)^{3}}\nabla v, $$
(22)

and, consequently, for \(v\in \dot {W}_{2}^{1}({\Omega })\cap {W_{1}^{2}}({\Omega })\),

$$ -({\Delta} v,\text{sign}(v)) = (\nabla v,\varepsilon^{2}\sigma_{\varepsilon}(v)^{-3}\nabla v) \ge 0. $$
(23)
  1. (i)

    First, we consider the case α>0. Subtracting the second from the first equation in (20) yields

    $$-\partial_{\tau}(G-H) - {\Delta}(G-H) + (\alpha+b)(G-H) = {\Delta} H. $$

    We multiply this by sign ε (G−H) and integrate the result over Ω. Further using the relations (22) and (23), we get

    $$-(\partial_{\tau}\sigma_{\varepsilon}(G-H),1) + ((\alpha+b)(G-H),\text{sign}_{\varepsilon}(G-H)) \le ({\Delta} H,\text{sign}_{\varepsilon}(G-H)). $$

    We integrate this with respect to time from τ to t, let ε→0, and observe that b≥0, to obtain that, for τ∈[0,t],

    $$\|(G-H)(\tau)\|_{1} + {\int}_{\tau}^{t} \|b(G-H)\|_{1}ds + \alpha{\int}_{\tau}^{t} \|G-H\|_{1}ds \le 1 + {\int}_{\tau}^{t} \|{\Delta} H\|_{1}ds. $$

    Using the identities

    $$\begin{array}{@{}rcl@{}} &&\partial_{\tau}{\Delta} H - \alpha{\Delta} H = -\alpha{\Delta} G,\quad {\Delta} H\vert_{\tau=t} = 0,\\ &&{\Delta} G = b(G-H)-\partial_{\tau} G,\quad \partial_{\tau} G = \partial_{\tau} H - \frac{1}{\alpha}\partial_{\tau}^{2}H, \end{array} $$

    we conclude that, for 0≤s≤t,

    $${\Delta} H(s) = {\alpha{\int}_{s}^{t}} e^{\alpha(s-r)}{\Delta} Gdr = {\alpha{\int}_{s}^{t}} e^{\alpha(s-r)}\left\{ b(G-H) - \partial_{r}H + \alpha^{-1}{\partial_{r}^{2}}H\right\}dr. $$

    Through integration by parts of the last term, the term ∂ r H cancels,

    $$\begin{array}{@{}rcl@{}} {\Delta} H(s) &=& {\alpha{\int}_{s}^{t}} e^{\alpha(s-r)}b(G-H)dr - \partial_{s}H(s) + e^{\alpha(s-t)}\partial_{s}H(t)\\ &=& {\alpha{\int}_{s}^{t}} e^{\alpha(s-r)}b(G-H)dr + \alpha(G-H)(s) - \alpha e^{\alpha(s-t)}\chi. \end{array} $$

    Taking norms and integrating with respect to time yields

    $${\int}_{\tau}^{t} \|{\Delta} H\|_{1}ds \le \alpha{\int}_{\tau}^{t}{{\int}_{s}^{t}} e^{\alpha(s-r)} \|b(G-H)\|_{1}drds + \alpha{\int}_{\tau}^{t} \|G-H\|_{1}ds + 1. $$

    Now, combining this estimate with (23) gives us

    $$ \|(G-H)(\tau)\|_{1} + {\int}_{\tau}^{t} \|b(G-H)\|_{1}ds \le 2 + {\int}_{\tau}^{t}{{\int}_{s}^{t}} \alpha e^{\alpha(s-r)}\|b(G-H)\|_{1}\,dr\,ds. $$
    (24)

    This estimate has the form

    $$ f(\tau) + {\int}_{\tau}^{t} g(s)ds \le \beta + {\int}_{\tau}^{t}{{\int}_{s}^{t}} \alpha e^{\alpha(s-r)}g(r)\,dr\,ds $$
    (25)

    for functions f∈C[0,t], g∈L 1(0,t), and constants β≥0, α≥0. The inequality (25) implies that

    $$ f(\tau) + \alpha{\int}_{\tau}^{t} f(s)ds + {\int}_{\tau}^{t} g(s)ds \le \beta(1+\alpha(t-\tau)),\quad \tau\in[0,t]. $$
    (26)

    This can be seen as follows. Define, for τ∈[0,t],

    $$w(\tau) := {\int}_{\tau}^{t}{{\int}_{s}^{t}} \alpha e^{\alpha(s-t)}g(r)\,dr\,ds. $$

    Then,

    $$w^{\prime}(\tau) = -{\int}_{\tau}^{t} \alpha e^{\alpha(\tau-r)}g(r)dr,\quad w^{\prime\prime}(\tau) = \alpha g(\tau) + \alpha w^{\prime}(\tau), $$

    and hence, as w(t) = w ′(t)=0,

    $${\int}_{\tau}^{t} g(s)\,ds = -\alpha^{-1} w^{\prime}(\tau) + w(\tau). $$

    Combining this inequality with (25) implies

    $$f(\tau) - \alpha^{-1} w^{\prime}(\tau) \le \beta,\quad \tau\in [0,t], $$

    and therefore, by integration,

    $$w(\tau) \le {\int}_{\tau}^{t} \alpha\{\beta - f(s)\}ds w(\tau) = \beta \alpha(t-\tau) - \alpha{\int}_{\tau}^{t} f(s)ds. $$

    Inserting this inequality into (25) concludes the proof of (26).

    Now, using the estimate (26), we conclude from (24) that

    $$\|(G-H)(\tau)\|_{1} + \alpha {\int}_{\tau}^{t} \|G-H\|_{1}ds + {\int}_{\tau}^{t} \|b(G-H)\|ds \le 2(1+\alpha(t-\tau)). $$

    This directly implies the first part of the asserted estimates (21), and then also the second part upon integration over time, for the case α>0.

  2. (ii)

    For α=0, equations (20) reduce to

    $$-\partial_{\tau} G - {\Delta} G + bG = 0 \quad \text{in}~ Q_{t},\quad G\vert_{\partial{\Omega}}=0,\quad G\vert_{\tau=t}=\chi. $$

    Then, multiplying with sign ε (G) leads us to

    $$\|G(\tau)\|_{1} + {\int}_{\tau}^{t} \|bG\|_{1}ds \le 1, $$

    i.e., in particular to (21). This completes the proof.

□

Next, we introduce the discrete analogues {G h ,H h }∈C 1([0,T];V h )×C 1([0,T];W h ) of the functions {G.H} as the solutions of the following problems, also running backward in time,

$$\begin{array}{@{}rcl@{}} -\partial_{\tau} G_{h} - {\Delta}_{h}G_{h} &=& -P_{h}b(G_{h}-H_{h})\quad\text{on}~ [0,t),\quad G_{h}(t) = P_{h}\chi, \\ -\partial_{\tau} H_{h} &=& \alpha(G_{h}-H_{h})\quad\text{in}~ [0,t),\quad H_{h}(t)=0. \end{array} $$

Using this notation, we find by a simple calculation that, with the total time derivative d τ ,

$$\begin{array}{@{}rcl@{}} d_{\tau}\left( (u,G)+(v,H)\right) &=& (\partial_{\tau} u,G) + (\partial_{\tau} v,H) + (u,\partial_{\tau} G) + (v,\partial_{\tau} H) \\ &=& (f+{\Delta} u +\alpha v - \phi(u),G) + (\phi(u)-\alpha v,H) \\ &&+ (u,b(G-H)-{\Delta} G) + (v,\alpha(H-G)) \\ &=& (f-\phi(u)+bu,G) + (\phi(u)-bu,H), \end{array} $$

and, analogously,

$$\begin{array}{@{}rcl@{}} d_{\tau}\left( (u_{h},G_{h})+(v_{h},H_{h})\right) &=& (\partial_{\tau} u_{h},G_{h}) + (\partial_{\tau} v_{h},H_{h}) + (u_{h},\partial_{\tau} G_{h}) + (v_{h},\partial_{\tau} H_{h}) \\ &=& (f+{\Delta}_{h} u_{h} +\alpha v_{h} - \phi(u_{h}),G_{h}) + (\phi(u_{h})-\alpha v_{h},H_{h}) \\ &&+ (u_{h},b(G_{h}-H_{h})-{\Delta}_{h} G_{h}) + (v_{h},\alpha(H_{h}-G_{h})) \\ &=& (f-\phi(u_{h})+bu_{h},G_{h}) + (\phi(u_{h})-bu_{h},H_{h}). \end{array} $$

Subtracting the above identities yields

$$\begin{array}{@{}rcl@{}} &&d_{\tau}\left( (u,G)-(u_{h},G_{h}) + (v,H)-(v_{h},H_{h})\right) \\ &&\quad= (f,G-G_{h}) - \phi(u)-b(u-u_{h}),G) \\ &&\qquad+ (\phi(u)-b(u-u_{h}),H) + (\phi(u_{h}),G_{h}-H_{h}) \\ &&\quad= (f,G-G_{h}) + (\phi(u)-\phi(u_{h})-b(u-u_{h}),H-G) \\ &&\qquad- (\phi(u_{h}),G-G_{h}) + (\phi(u_{h}),H-H_{h}). \end{array} $$

We integrate in this identity with respect to time and use the inequality (18) to obtain

$$\begin{array}{@{}rcl@{}} ((u-u_{h})(t),\chi) &\le& (u^{0},(G-G_{h})(0) + (v^{0},(H-H_{h})(0)) \\ &&+ {{\int}_{0}^{t}}\left\{(f,G-G_{h}) - (\phi(u_{h}),G-G_{h}) + (\phi(u_{h}),H-H_{h})\right\}d\tau \\ &&+ \frac{h^{2}}{t}{{\int}_{0}^{t}} \|H-G\|_{1}\tau. \end{array} $$

Then, in virtue of the bounds

$$\max\left\{\|u^{0}\|_{2,\infty},\|v^{0}\|_{\infty}, \|f\|_{\infty;Q_{T}} \right\} \le K,\quad \|\phi(u_{h})\|_{\infty;Q_{t}} \le M, $$

and Lemma 1, we can conclude that

$$\begin{array}{@{}rcl@{}} |((u-u_{h})(t),\chi)| &\le& K\left\{\|(G-G_{h})(0)\|_{-2,1} + \|(H-H_{h})(0)\|_{1}\right\}\\ &&+ (K+M)\|G-G_{h}\|_{1;Q_{t}} + M\|H-H_{h}\|_{1;Q_{t}} \\ &&+ 2(1+\alpha t)h^{2}, \end{array} $$

where ∥⋅∥−2,1 denotes the norm of the dual space of \(W_{\infty }^{2}({\Omega })\cap \dot {W}_{2}^{1}({\Omega })\). Now, the next lemma implies the desired estimate (19), concluding the proof of the theorem.

Lemma 2

There hold the estimates

$$\begin{array}{@{}rcl@{}} \|(G-G_{h})(0)\|_{-2,1} + \|G-G_{h}\|_{1;Q_{t}} &\le& c(1+\alpha t) h^{2}|\log(h)|^{2}, \end{array} $$
(27)
$$\begin{array}{@{}rcl@{}} \alpha^{-1}\|(H-H_{h})(0)\|_{1} + \|H-H_{h}\|_{1;Q_{t}} &\le& c(1+\alpha t) h^{2}|\log(h)|^{2}, \end{array} $$
(28)

for α>0, where the constant c is particularly independent of the constants α,K,M, of t and of the solution {u,v}.

Proof

The proof is by a parabolic duality argument. With arbitrarily fixed g∈L ∞ (Q t ) and \(z^{0}\in W_{\infty }^{2}({\Omega })\cap \dot {W}_{2}^{1}({\Omega })\), we consider the auxiliary problem

$$\partial_{\tau} z - {\Delta} z = g\quad\text{in}~ Q_{t},\quad z\vert_{\partial{\Omega}}=0,\quad z\vert_{\tau=0}=z^{0}, $$

and its discrete analogues in V h ,

$$ \partial_{\tau} z_{h} - {\Delta}_{h} z_{h} = P_{h}g\quad\text{in}~ Q_{t},\quad z_{h}\vert_{\partial{\Omega}}=0,\quad z_{h}\vert_{\tau=0}=P_{h}z^{0}. $$

Then, for τ∈(0,t) there holds

$$\begin{array}{@{}rcl@{}} (G-G_{h},g) &=& (G-G_{h},\partial_{\tau} z-{\Delta} z) \\ &=& d_{\tau}(G-G_{h},z) - (\partial_{\tau}(G-G_{h}),z) + (\nabla(G-G_{h}),\nabla z), \end{array} $$

and, using the orthogonality relations for G−G h and z−z h ,

$$\begin{array}{@{}rcl@{}} (G-G_{h},g) &=& d_{\tau}(G-G_{h},z) - (\partial_{\tau}(G-G_{h}),z-z_{h}) + (\nabla(G-G_{h}),\nabla(z-z_{h}))\\ &=& d_{\tau}(G-G_{h},z_{h}) + (G-G_{h},\partial_{\tau}(z-z_{h})) + (\nabla(G-G_{h}),\nabla(z-z_{h})) \\ &=& d_{\tau}(G-G_{h},z_{h}) + (G,\partial_{\tau}(z-Z_{h})) + (\nabla G,\nabla(z-z_{h})) \\ &=& d_{\tau}(G-G_{h},z_{h}) + d_{\tau}(G,z-z_{h}) - (\partial_{\tau} G,z-z_{h}) + (\nabla G,\nabla(z-z_{h})) \\ &=& d_{\tau}\left\{(G,z)-(G_{h},z_{h})\right\} - (b(G-H),z-z_{h}). \end{array} $$

Integrating this over [0,T], and observing the initial conditions for G and z, yields

$$((G-G_{h})(0),z^{0}) + {{\int}_{0}^{t}} (G-G_{h},g)d\tau = (\chi,(z-z_{h})(t)) - {{\int}_{0}^{t}} (b(G-H),z-z_{h})d\tau, $$

and, consequently, using the bound for \(\|b(G-H)\|_{1;Q_{t}}\) from Lemma 1,

$$((G-G_{h})(0),z^{0}) + {{\int}_{0}^{t}} (G-G_{h},g)d\tau \le (3+2\alpha t)\|z-z_{h}\|_{\infty;Q_{t}}. $$

From Schatz et al. [17], Rannacher [15], and particularly from Chen [3], we recall the error estimate

$$\|z-z_{h}\|_{\infty;Q_{t}} \le ch^{2}|\log(h)|^{2}\left\{\|z^{0}\|_{2,\infty} + \|g\|_{\infty;Q_{t}}\right\}, $$

which holds uniformly for t≥0. This implies the desired estimate (27).

To prove (28), we multiply the error identity

$$-\partial_{\tau}(H-H_{h}) + \alpha(H-H_{h}) = \alpha(G-G_{h}) $$

by sign ε (H−H h ), integrate with respect to space and time and obtain, after letting ε→0,

$$\|(H-H_{h})(0)\|_{1} + \alpha\|H-H_{h}\|_{1;Q_{t}} \le \alpha\|G-G_{h}\|_{1;Q_{t}}. $$

Then, the desired estimate follows from (28). □

5 Numerical Results

In the following, we present some test calculations for the model problem (6),

$$\begin{array}{@{}rcl@{}} \partial_{t}u - {\Delta} u + \partial_{t}v &=& f \quad \text{in}~ Q_{T},\quad u\vert_{\partial{\Omega}}=0,\quad u\vert_{t=0} = u^{0},\\ \partial_{t}v + \alpha v &=& \phi(u)\quad\text{in}~ Q_{T},\quad v\vert_{t=0} = v^{0}, \end{array} $$
(29)

on Q T = B 1(0)×[0,1], in order to examine the theoretical results of this paper. The spatial domain is the unit circle \({\Omega } = B_{1}(0)\subset \mathbb R^{2}\). We choose the prototypical nonlinearity ϕ(s):=sgn(s)|s|p, for 0<p≤1. The results of this section are taken from the diploma thesis Lambrecht [11]. The computations have been done using the finite element software “deal.II” [5].

5.1 Test with Known Solution

By this first test, we mainly want to validate the implemented solution method. We choose the nonlinearity with p=1/3 and the parameter value α=1. The prescribed “exact” solution is

$$\begin{array}{@{}rcl@{}} u(x,y,t) &=& e^{-3t}\sin(r(x,y)), \\ v(x,y,t) &=& te^{-t}\sin^{1/3}(r(x,y)), \\ r(x,y) &=& \tfrac{10}{9}\pi (x^{2} + y^{2}) - \tfrac{1}{9} \pi. \end{array} $$

The corresponding forcing term has the form

$$\begin{array}{@{}rcl@{}} f(x,y,t) &=& \partial_{t}u(x,y,t) - {\Delta} u(x,y,t) + \partial_{t}v(x,y,t) \\ &=& -3e^{-3t}\sin(r(x,y))-4e^{-3t}(\tfrac{10}{9}\pi)\cos(r(x,y)) \\ &&+ e^{-3t}(\tfrac{20}{9}\pi)^{2}(x^{2} + y^{2})\sin(r(x,y))+ e^{-t}\sin^{1/3}(r(x,y))\\ &&- te^{-t}\sin^{1/3}(r(x,y)). \end{array} $$

The u- and v-component of this exact solution at final time t=1, are shown in Fig. 1. As we compare the computed solution to this exact solution, the time step size is chosen sufficiently small, k=2−9, such that the total error is dominated by the spatial discretization error. The observed error behavior under mesh refinement is shown in Table 1. We see that the theoretically predicted orders of convergence are essentially confirmed.

Fig. 1
figure 1

Case p=1/3: u-component (left) and v-component (right) of exact (top) and computed (bottom) solution at time t=1.0 on 1280 mesh cells using 32 time steps

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Table 1 Maximum norm errors on a sequence of uniformly refined spatial meshes for α=1 and p=1/3 (case of known solution)
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5.2 Test with Unknown Solution

Next, we consider the model problem (29) on the space-time region Q T = B 1(0)×[0,1] with the data f≡−100, α=0.5, u| t=0≡0, u| ∂Ω≡25. In this case, the exact solution is not known and has to be approximated by a prior computation on a very fine space-time mesh. In the test computation, the rather coarse time step size k=0.025 is used. This is justified as only the intrinsic order of convergence of the spatial discretization is measured. By this test, we want to check the predicted orders of convergence depending on the parameter p, i.e., the Hölder continuity of the nonlinearity ϕ(s)=sgn(s)|s|p. The corresponding results are shown in Figs. 2 and 3 and in Tables 2, 3, 4, 5, 6, and 7. We see that the predicted optimal order of L ∞-convergence for the variable u is well confirmed, while that predicted in the literature for the L 2-convergence is only suboptimal. The observed order of L ∞-convergence for the variable v seems to be rather 1 + p than the predicted suboptimal order 2p. The clarification of this discrepancy is left as an open problem.

Fig. 2
figure 2

Case p=1/3: u-component (left) and v-component (right) of the computed solution at times t=0.5 (top) and t=1.0 (bottom) on 1280 mesh cells using 32 time steps

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Fig. 3
figure 3

Case p=1/6: u-component (left) and v-component (right) of the computed solution at times t=0.5 (top) and t=1.0 (bottom) on 1280 mesh cells using 32 time steps

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Table 2 Error behavior at time t=1 for p=1/3
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Table 3 Error behavior at time t=1 for p=1/6
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Table 4 Error behavior at time t=1 for p=1/2
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Table 5 Error behavior at time t=1 for p=2/3
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Table 6 Error behavior at time t=1 for p=5/6
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Table 7 Error behavior at time t=1 for p=1
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6 Conclusion

Our analysis of the spatial semi-discretization of a degenerate parabolic system with a monotone but non-Lipschitzian nonlinearity provides an optimal-order L ∞-error estimate for the first solution component, u, which does not depend on the Hölder exponent p∈(0,1] of the nonlinearity. The occurrence of the term |log(h)2| in the error estimate seems natural in the case of low-order finite elements as it is known to be unavoidable even in the simpler situations of the Poisson or heat equation. This second-order convergence is well confirmed by the numerical tests. For the second solution component, v, we can derive only an estimate of the reduced order \(\mathcal {O}(h^{2p}|\log (h)|^{2p})\), depending explicitly on the Hölder exponent p of the nonlinearity. In view of our numerical results, this predicted order seems optimal in general but only suboptimal in certain special cases. The clarification of this discrepancy is left as an open problem.