Constructive error analysis of a full-discrete finite element method for the heat equation

Abstract

We present a new full-discrete finite element method for the heat equation, and show the numerical stability of the method by verified computations. Since, in the error analysis, we use the constructive error estimates proposed by Nakao et al. (SIAM J Numer Anal 51(3):1525–1541, 2013) this work is considered as an extension of that paper. We emphasize that the concerned scheme seems to use the quite standard Galerkin method and is easy to implement for evolutionary equations compared with previous ones. In the constructive error estimates, we effectively use the numerical computations with guaranteed accuracy.

1 Introduction

The purpose of this paper is to establish the constructive a priori error estimates for a full-discrete approximations $Q_h^k u$, which is defined in Sect. 3, of the solution u to the following linear heat equation:

$$\begin{aligned} \begin{array}{rclcl} {\frac{\partial {}}{\partial {t}}}u-\nu \varDelta u&{}=&{}f&{}{\mathrm{in}}&{}\varOmega \times J,\\ u(x,t)&{}=&{}0&{}{\mathrm{on}}&{}\partial \varOmega \times J,\\ u(0)&{}=&{}0&{}{\mathrm{in}}&{}\varOmega . \end{array} \end{aligned}$$

(1)

Here, $\varOmega \subset \mathbb {R}^d$$(d \in \{1,2,3\})$ is a bounded polygonal or polyhedral domain; $J:=(0,T) \subset \mathbb {R}$, (for a fixed $T<\infty$) is a bounded open interval; the diffusion coefficient $\nu$ is a positive constant; and $f \in L^2\bigl (J;L^2(\varOmega )\bigr )$, where, in general for any normed space Y, we define the time-dependent Lebesgue space $L^2\bigl (J;Y \bigr )$ as a space of square integrable Y-valued functions on J. That is,

$$\begin{aligned} f \in L^2\bigl (J;Y \bigr ) \Leftrightarrow \int _J ||f(t)||_Y^2 dt < \infty , \end{aligned}$$

Moreover, we define the inner product $\left<f,g\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}:=\left<f,g\right>_{L^2(\varOmega \times J)}$ for $f,g\in L^2\bigl (J;L^2(\varOmega )\bigr )$.

We know that the Eq. (1) has a unique solution (see, e.g., [8]). It is shown a full-discrete approximation $\hat{Q}_h^k u$, which is defined in Sect. 3, of the solution u for the heat Eq. (1) in [3]. Then, the projection $\hat{Q}_h^k$ satisfies $L^2H^1_0$-stability, however, is not $V^1$-stable. Thus, our purpose of this paper is to propose a new projection $Q_h^k$ which satisfies $V^1$-stability as well as it has $L^2H^1_0$-stability, and we show the constructive a priori error estimates for the projection $Q_h^k$.

In the discussion below, we refer to the a priori estimates as ‘constructive’ if all the constants can be numerically determined.

2 Notations and preliminary

The notations to the spaces in this paper are very similar to that presented in [5], we include these here for the sake of convenience.

We denote $L^2(\varOmega )$ and $H^1(\varOmega )$ as the usual Lebesgue and the first order $L^2$-Sobolev spaces on $\varOmega$, respectively, and by $\left<u,v\right>_{L^2(\varOmega )}:=\int _\varOmega u(x)v(x)\,dx$ the natural inner product for u, $v\in L^2(\varOmega )$. By considering the boundary and initial conditions, we define the following subspaces of $H^1(\varOmega )$ and $H^1(J)$ as

$$\begin{aligned} H_0^1(\varOmega ):={\left\{ {u \in H^1(\varOmega )}~;~{u = 0 \text { on } \partial \varOmega }\right\} } \quad \text{ and } \quad V^1(J):={\left\{ {u \in H^1(J)}~;~{u(0) = 0}\right\} }, \end{aligned}$$

respectively. These are Hilbert spaces with inner products

$$\begin{aligned} \left<u,v\right>_{H_0^1(\varOmega )}:=\left<\nabla u,\nabla v\right>_{L^2(\varOmega )^d}\quad \text{ and } \quad \left<u,v\right>_{V^1(J)}:=\left<{\frac{\partial {u}}{\partial {t}}},{\frac{\partial {v}}{\partial {t}}}\right>_{L^2(J)}. \end{aligned}$$

Let $X(\varOmega )$ be a subspace of $L^2(\varOmega )$ defined by $X(\varOmega ):={\left\{ {u \in L^2(\varOmega )}~;~{\triangle u \in L^2(\varOmega )}\right\} }$. We define

$$\begin{aligned} V^1\bigl (J;L^2(\varOmega )\bigr ):={\left\{ {u \in L^2\bigl (J;L^2(\varOmega )\bigr )}~;~{{\frac{\partial {u}}{\partial {t}}} \in L^2\bigl (J;L^2(\varOmega )\bigr ),\ u(x,0)=0 \quad \text { in }\quad \varOmega }\right\} }, \end{aligned}$$

with inner product $\left<u,v\right>_{V^1\bigl (J;L^2(\varOmega )\bigr )}:=\left<{\frac{\partial {u}}{\partial {t}}},{\frac{\partial {v}}{\partial {t}}}\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}$. In the following discussion, abbreviations like $L^2H_0^1$ for $L^2\bigl (J;H_0^1(\varOmega )\bigr )$ will often be used. We set $V(\varOmega ,J):= V^1\bigl (J;L^2(\varOmega )\bigr ) \cap L^2\bigl (J;H_0^1(\varOmega )\bigr )$. Moreover, we denote the partial differential operator $\triangle _t:V(\varOmega ,J) \cap L^2\bigl (J;X(\varOmega )\bigr ) \rightarrow L^2\bigl (J;L^2(\varOmega )\bigr )$ by $\triangle _t:={\frac{\partial {}}{\partial {t}}}-\nu \triangle$.

Now let $S_h(\varOmega )$ be a finite-dimensional subspace of $H_0^1(\varOmega )$ dependent on the parameter h. For example, $S_h(\varOmega )$ is considered to be a finite element space with mesh size h. Let n be the degree of freedom for $S_h(\varOmega )$, and let $\{\phi _i\}_{i=1}^n \subset H_0^1(\varOmega )$ be the basis of $S_h(\varOmega )$. Similarly, let $V_k^1(J)$ be an approximation subspace of $V^1(J)$ dependent on the parameter k. Let m be the degree of freedom for $V_k^1(J)$, and let $\{\psi _i\}_{i=1}^m \subset V_k^1(J)$ be the basis of $V_k^1(J)$. Let $V^1\bigl (J;S_h(\varOmega )\bigr )$ be a subspace of $V(\varOmega ,J)$ corresponding to the semidiscretized approximation in the spatial direction. We define the $H_0^1$-projection $P_h^1 u \in S_h(\varOmega )$ of any element $u \in H_0^1(\varOmega )$ by the following variational equation:

$$\begin{aligned} \left<\nabla (u-P_h^1u),\nabla v_h\right>_{L^2(\varOmega )^d} = 0, \quad \forall v_h \in S_h(\varOmega ). \end{aligned}$$

(2)

Similarly, for any element $u \in V^1(J)$, the $V^1$-projection $P^k_1:V^1(J) \rightarrow V_k^1(J)$ is defined by follows:

$$\begin{aligned} \left<{\frac{\partial {}}{\partial {t}}}(u-P_1^ku),{\frac{\partial {}}{\partial {t}}} v_k\right>_{L^2(J)} = 0, \quad \forall v_k \in V_k^1(J). \end{aligned}$$

Now let $\varPi _k:V^1(J) \rightarrow V_k^1(J)$ be an interpolation operator. Namely, if the nodal points of J are given by $0=t_0<t_1<\cdots <t_m=T$, then for an arbitrary $u \in V^1(J)$, the interpolation $\varPi _ku$ is defined as the function in $V_k^1(J)$ satisfying:

$$\begin{aligned} u(t_i) = \bigl (\varPi _ku\bigr )(t_i), \quad \forall i \in \{1,\ldots ,m\}. \end{aligned}$$

(3)

We know that there exist constants $C_\varOmega (h)>0$, $C_J(k)>0$ and $C_{{\text {inv}}}(h)>0$ satisfying

$$\begin{aligned} \left\| u-P_h^1u\right\| _{H_0^1(\varOmega )}\le & {} C_\varOmega (h)\left\| \triangle u\right\| _{L^2(\varOmega )}, \quad \forall u\in H_0^1(\varOmega ) \cap X(\varOmega ), \\ \left\| u-\varPi _ku\right\| _{L^2(J)}\le & {} C_J(k)\left\| u\right\| _{V^1(J)}, \quad \forall u\in V^1(J),\\ \left\| u_h\right\| _{H_0^1(\varOmega )}\le & {} C_{{\text {inv}}}(h)\left\| u_h\right\| _{L^2(\varOmega )}, \quad \forall u_h\in S_h(\varOmega ). \end{aligned}$$

Moreover, there exists a Poincaré constant $C_p>0$ satisfying

$$\begin{aligned} \left\| u\right\| _{L^2(\varOmega )}\le & {} C_p\left\| u\right\| _{H^1_0(\varOmega )}, \quad \forall u\in H^1_0(\varOmega ). \end{aligned}$$

For example, if $\varOmega$ is a bounded open rectangular domain in $\mathbb {R}^d$, and $S_h(\varOmega )$ is the piecewise linear (P1) finite element space, then they can be taken by $C_\varOmega (h)=\frac{h}{\pi }$ (see, e.g., [4]) and $C_{{\text {inv}}}(h)=\frac{\sqrt{12}}{h_{\min }}$, where $h_{\min }$ is the minimum mesh size for $\varOmega$ (see, e.g., [7, Theorem 1.5]). Moreover, if $V_k^1(J)$ is the P1-finite element space, then it can be taken by $C_J(k)=\frac{k}{\pi }$ (see, e.g., [7, Theorem 2.4]).

From [1, Lemma 2.2], if $V_k^1(J)$ is P1-finite element space (i.e., the basis functions $\psi _i$ are piecewise linear functions), then $P^k_1$ coincides with $\varPi _k$. For any element $u \in V(\varOmega ,J)$, we define the semidiscrete projection $P_hu \in V^1\bigl (J;S_h(\varOmega )\bigr )$ by the following weak form:

$$\begin{aligned}&\left<{\frac{\partial {}}{\partial {t}}}\bigl (u(t)-P_hu(t)\bigr ),v_h\right>_{L^2(\varOmega )} + \nu \left<\nabla \bigl (u(t)-P_hu(t)\bigr ),\nabla v_h\right>_{L^2(\varOmega )^d} = 0, \nonumber \\&\quad \forall v_h \in S_h(\varOmega ),~~{\text {a.e.}}~~t \in J, \end{aligned}$$

(4)

where ${\text {a.e.}}~~$means an abbreviation for ’almost everywhere’.

Finally, the space $S_h^k(\varOmega ,J)$ is defined as the tensor product $V_k^1(J) \otimes S_h(\varOmega )$, which corresponds to a full discretization. Moreover, we define the full-discretization operator $P_h^k:V(\varOmega ,J) \rightarrow S_h^k(\varOmega ,J)$ by $P_h^k:=\varPi _kP_h$. In addition, we denote the matrix norm induced from the Euclidean 2-norm by $\Vert \cdot \Vert _{E}$ and denote the transposed matrix of the matrix $\mathbf{X}$ by $\mathbf{X}^{\mathrm{T}}$.

We show known results for the Eq. (1) below.

Theorem 1

(see Theorem 5.5, 5.6, and proof of Theorem 4.6 in [5]) Let $u\in V(\varOmega ,J) \cap L^2\bigl (J;X(\varOmega )\bigr )$ be a solution of (1) for $f\in L^2\bigl (J;X(\varOmega )\bigr )$. Then, we have the following estimations.

$$\begin{aligned} \left\| u-P_h^ku\right\| _{L^2\bigl (J;H_0^1(\varOmega )\bigr )}\le & {} C_1(h,k)\left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}, \end{aligned}$$

(5)

$$\begin{aligned} \left\| u-P_h^ku\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}\le & {} C_0(h,k)\left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )},\end{aligned}$$

(6)

$$\begin{aligned} \left\| u(T)-P_hu(T)\right\| _{L^2(\varOmega )}\le & {} c_0(h)\left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}, \end{aligned}$$

(7)

where $C_1(h,k):=\frac{2}{\nu }C_\varOmega (h) + C_{{\text {inv}}}(h)C_J(k)$, $C_0(h,k)=\frac{8}{\nu }C_\varOmega (h)^2+C_J(k)$ and $c_0(h)=\sqrt{\frac{8}{\nu }}C_\varOmega (h)$.

Note that the scheme in [5] is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. Since, in the derivation procedure, it uses the fundamental matrix of solution for ODEs associated with the semidiscrete approximation, it is necessary to implement the complicated verified computations on matrix functions to compute the approximation $P_h^ku$.

3 The full-discrete finite element method

We define the bi-linear form $a_0(\cdot ,\cdot )$ by

$$\begin{aligned} a_0(\phi ,\psi ):= & {} \left<{\frac{\partial {}}{\partial {t}}}\phi ,{\frac{\partial {}}{\partial {t}}}\psi \right>_{L^2\bigl (J;L^2(\varOmega )\bigr )} +\nu \left<\nabla \phi ,{\frac{\partial {}}{\partial {t}}}\nabla \psi \right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d}, \end{aligned}$$

for $\phi , \psi \in V(\varOmega ,J)$. Then, for any element $u \in V(\varOmega ,J)$, we define the full-discrete projection $Q_h^ku \in S_h^k(\varOmega ,J)$ by the following weak form:

$$\begin{aligned} a_0(u-Q_h^ku,v_h^k) = 0, \quad \forall v_h^k \in S_h^k(\varOmega ,J). \end{aligned}$$

Then the projection $Q_h^ku \in S_h^k(\varOmega ,J)$ for the solution of (1) satisfies the following weak form:

$$\begin{aligned} a_0(Q_h^ku,v_h^k) = \left<f,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}, \quad \forall v_h^k \in S_h^k(\varOmega ,J). \end{aligned}$$

(8)

Note that the present scheme by (8) need not any the complicated procedures like $P_h^k$ at all. We now have the following estimation from above definition.

Lemma 1

Let $u\in V(\varOmega ,J) \cap L^2\bigl (J;X(\varOmega )\bigr )$ be a solution of (1) for $f\in L^2\bigl (J;L^2(\varOmega )\bigr )$. Then, the full-discrete projection $Q_h^ku \in S_h^k(\varOmega ,J)$ satisfies the following $V^1$-stability:

$$\begin{aligned} \left\| Q_h^ku\right\| _{V^1\bigl (J;L^2(\varOmega )\bigr )}\le & {} \left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}. \end{aligned}$$

Proof

From (8) and $Q_h^ku(x,0)=0$ in $\varOmega$, if we take $v_h^k=Q_h^ku$ then it follows that

$$\begin{aligned} \left\| Q_h^ku\right\| _{V^1\bigl (J;L^2(\varOmega )\bigr )}^2 + \frac{\nu }{2}\left\| Q_h^ku(T)\right\| _{H^1_0(\varOmega )}^2= & {} a_0(Q_h^ku,Q_h^ku)\\= & {} \left<f,{\frac{\partial {}}{\partial {t}}}Q_h^ku\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}\\\le & {} \left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}\left\| Q_h^ku\right\| _{V^1\bigl (J;L^2(\varOmega )\bigr )}. \end{aligned}$$

Therefore, the proof is completed. $\square$

Next, we consider the estimation $\left\| Q_h^ku\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}$ for $u \in V(\varOmega ,J) \cap L^2\bigl (J;X(\varOmega )\bigr )$. We now define $\alpha _h^k\in S_h^k(\varOmega ,J)$ satisfying $\left<{\frac{\partial {}}{\partial {t}}}\alpha _h^k,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}=\left<f,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}$ for all $v_h^k \in S_h^k(\varOmega ,J)$. Note that by taking $v_h^k=\alpha _h^k$, it follows that $\left\| \alpha _h^k\right\| _{V^1\bigl (J;L^2(\varOmega )\bigr )}\le \left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}$, which also implies the unique existence of $\alpha _h^k$. We now define the matrices $\mathbf{A}$ and $\mathbf{M}$ in $\mathbb {R}^{mn \times mn}$ by

$$\begin{aligned} \mathbf{A}_{i,j}:= & {} \left<{\frac{\partial {}}{\partial {t}}}\varphi _j,{\frac{\partial {}}{\partial {t}}}\varphi _i\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )},\quad \mathbf{M}_{i,j}:=\left<\nabla \varphi _j,\nabla \varphi _i\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d},\\&\forall i,j \in \{1,\ldots ,mn\}, \end{aligned}$$

respectively. Since matrices $\mathbf{A}$ and $\mathbf{M}$ are symmetric and positive definite, we can denote the Cholesky decomposition as $\mathbf{A}=\mathbf{A}^{{\frac{1}{2}}}{} \mathbf{A}^{\frac{\mathrm{T}}{2}}$ and $\mathbf{M}=\mathbf{M}^{{\frac{1}{2}}}{} \mathbf{M}^{\frac{\mathrm{T}}{2}}$, respectively. Moreover, we define the matrix $\mathbf{B}$ in $\mathbb {R}^{mn \times mn}$ by

$$\begin{aligned} \mathbf{B}_{i,j}:=\left<\nabla \varphi _j,{\frac{\partial {}}{\partial {t}}}\nabla \varphi _i\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d},\quad \forall i,j \in \{1,\ldots ,mn\}. \end{aligned}$$

From the fact that $Q_h^ku$ and $\alpha _h^k$ belong to $S_h^k(\varOmega ,J)$, there exist coefficient vectors $\mathfrak {u}:=(\mathfrak {u}_1,\ldots ,\mathfrak {u}_{mn})^{\mathrm{T}}$ and $\mathfrak {a}:=(\mathfrak {a}_1,\ldots ,\mathfrak {a}_{mn})^{\mathrm{T}}$ in $\mathbb {R}^{mn}$ such that $Q_h^ku= \sum _{i=1}^{mn} \mathfrak {u}_i\varphi _i = \varphi ^{\mathrm{T}}\mathfrak {u}$ and $\alpha _h^k= \sum _{i=1}^{mn} \mathfrak {a}_i\varphi _i = \varphi ^{\mathrm{T}}\mathfrak {a}$ where $\varphi := (\varphi _1,\ldots ,\varphi _{mn})^{\mathrm{T}}$. Then, the equation (8) is equivalent to the following.

$$\begin{aligned} (\mathbf{A}+\nu \mathbf{B})\mathfrak {u}=\mathbf{A}\mathfrak {a}. \end{aligned}$$

(9)

Thus we have the following result.

Lemma 2

Let $u\in V(\varOmega ,J) \cap L^2\bigl (J;X(\varOmega )\bigr )$ be a solution of (1) for $f\in L^2\bigl (J;L^2(\varOmega )\bigr )$. Then, the full-discrete projection $Q_h^ku \in S_h^k(\varOmega ,J)$ is bounded as

$$\begin{aligned} \left\| Q_h^ku\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )} \le \eta \left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}, \end{aligned}$$

where $\eta :=\Vert \mathbf{M}^{\frac{\mathrm{T}}{2}}(\mathbf{A}+\nu \mathbf{B})^{-1}{} \mathbf{A}^{\frac{1}{2}}\Vert _E$.

Proof

From (9), we obtain

$$\begin{aligned} \left\| Q_h^ku\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}= & {} \Vert \mathbf{M}^{\frac{\mathrm{T}}{2}}\mathfrak {u}\Vert _E =\Vert \mathbf{M}^{\frac{\mathrm{T}}{2}}(\mathbf{A}+\nu \mathbf{B})^{-1}{} \mathbf{A}\mathfrak {a}\Vert _E\\\le & {} \Vert \mathbf{M}^{\frac{\mathrm{T}}{2}}(\mathbf{A}+\nu \mathbf{B})^{-1}\mathbf{A}^{\frac{1}{2}}\Vert _E\Vert \mathbf{A}^{\frac{\mathrm{T}}{2}}\mathfrak {a}\Vert _E\\= & {} \eta \left\| \alpha _h^k\right\| _{V^1\bigl (J;L^2(\varOmega )\bigr )}\\\le & {} \eta \left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}. \end{aligned}$$

Therefore, the proof is completed. $\square$

Now, in order to compare our scheme with another one presented in [3], we give some arguments below. By using $S_h^k(\varOmega ,J)$, we define the rather simple and natural looking full-discrete finite element scheme in both directions for the problem (1). First, we define the bi-linear form $\hat{a}_0(\cdot ,\cdot )$ by

$$\begin{aligned} \hat{a}_0(\phi ,\psi ):= & {} \left<{\frac{\partial {}}{\partial {t}}}\phi ,\psi \right>_{L^2\bigl (J;L^2(\varOmega )\bigr )} +\nu \left<\nabla \phi ,\nabla \psi \right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d}, \end{aligned}$$

for $\phi , \psi \in V(\varOmega ,J)$. Then, for any element $u \in V(\varOmega ,J)$, we define the full-discrete projection $\hat{Q}_h^ku \in S_h^k(\varOmega ,J)$ by the following weak form:

$$\begin{aligned} \hat{a}_0\left( u-\hat{Q}_h^ku,v_h^k\right) = 0, \quad \forall v_h^k \in S_h^k(\varOmega ,J). \end{aligned}$$

Using above, the projection $\hat{Q}_h^ku \in S_h^k(\varOmega ,J)$ for the solution of (1) satisfies the following weak form:

$$\begin{aligned} \hat{a}_0\left( \hat{Q}_h^ku,v_h^k\right) = \left<f,v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}, \quad \forall v_h^k \in S_h^k(\varOmega ,J). \end{aligned}$$

(10)

Let $u\in V(\varOmega ,J) \cap L^2\bigl (J;X(\varOmega )\bigr )$ be a solution of (1) for $f\in L^2\bigl (J;L^2(\varOmega )\bigr )$ Then, by taking $v_h^k=\hat{Q}_h^ku$, we can obtain

$$\begin{aligned} \frac{1}{2}\left\| \hat{Q}_h^ku(T)\right\| _{L^2(\varOmega )}^2 + \nu \left\| \hat{Q}_h^ku\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}^2= & {} \hat{a}_0(\hat{Q}_h^ku,\hat{Q}_h^ku)\\= & {} \left<f,\hat{Q}_h^ku\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}\\\le & {} \left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}\left\| \hat{Q}_h^ku\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}\\\le & {} C_p\left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}\left\| \hat{Q}_h^ku\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}. \end{aligned}$$

Thus we have the following $L^2H^1_0$-stability:

$$\begin{aligned} \left\| \hat{Q}_h^ku\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )} \le \frac{C_p}{\nu }\left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}. \end{aligned}$$

Moreover, we consider the estimation $\left\| \hat{Q}_h^ku\right\| _{V^1\bigl (J;L^2(\varOmega )\bigr )}$ for $u \in V(\varOmega ,J) \cap L^2\bigl (J;X(\varOmega )\bigr )$. Now we define $\hat{\alpha }_h^k\in S_h^k(\varOmega ,J)$ satisfying $\left<\hat{\alpha }_h^k,v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}=\left<f,v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}$ for all $v_h^k \in S_h^k(\varOmega ,J)$. Note that by taking $v_h^k=\hat{\alpha }_h^k$, it follows that $\left\| \hat{\alpha }_h^k\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}\le \left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}$. Moreover, from (10), we obtain

$$\begin{aligned} \hat{a}_0\left( \hat{Q}_h^ku,v_h^k\right) =\left<\hat{\alpha }_h^k,v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}. \end{aligned}$$

(11)

We now define the matrices $\mathbf{G}$ and $\mathbf{U}$ in $\mathbb {R}^{mn \times mn}$ by

$$\begin{aligned} \mathbf{G}_{i,j}:= & {} \left<{\frac{\partial {}}{\partial {t}}}\varphi _j,\varphi _i\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )},\quad \mathbf{U}_{i,j}:=\left<\varphi _j,\varphi _i\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d},\\&\forall i,j \in \{1,\ldots ,mn\}, \end{aligned}$$

respectively. Since the matrix $\mathbf{U}$ is symmetric and positive definite, we denote the Cholesky decomposition as $\mathbf{U}=\mathbf{U}^{\frac{1}{2}}{} \mathbf{U}^{\frac{\mathrm{T}}{2}}$. From the fact that $\hat{Q}_h^ku$ and $\hat{\alpha }_h^k$ in $S_h^k(\varOmega ,J)$, there exist coefficient vectors $\hat{\mathfrak {u}}:=(\hat{\mathfrak {u}}_1,\ldots ,\hat{\mathfrak {u}}_{mn})^{\mathrm{T}}$ and $\hat{\mathfrak {a}}:=(\hat{\mathfrak {a}}_1,\ldots ,\hat{\mathfrak {a}}_{mn})^{\mathrm{T}}$ in $\mathbb {R}^{mn}$ such that $\hat{Q}_h^ku= \sum _{i=1}^{mn} \hat{\mathfrak {u}}_i\varphi _i = \varphi ^{\mathrm{T}}\hat{\mathfrak {u}}$ and $\hat{\alpha }_h^k= \sum _{i=1}^{mn} \hat{\mathfrak {a}}_i\varphi _i = \varphi ^{\mathrm{T}}\hat{\mathfrak {a}}$, respectively. Then, the variational Eq. (11) is equivalent to the following.

$$\begin{aligned} (\mathbf{G}+\nu \mathbf{M})\hat{\mathfrak {u}}=\mathbf{U}\hat{\alpha }. \end{aligned}$$

Letting $\hat{\eta }:=\Vert \mathbf{A}^{\frac{\mathrm{T}}{2}}(\mathbf{G}+\nu \mathbf{M})^{-1}{} \mathbf{U}^{\frac{1}{2}}\Vert _E$, it follows that

$$\begin{aligned} \left\| \hat{Q}_h^ku\right\| _{V^1\bigl (J;L^2(\varOmega )\bigr )}=\Vert \mathbf{A}^{\frac{\mathrm{T}}{2}}\hat{\mathfrak {u}}\Vert _E= & {} \Vert \mathbf{A}^{\frac{\mathrm{T}}{2}}(\mathbf{G}+\nu \mathbf{M})^{-1}{} \mathbf{U}\hat{\mathfrak {a}}\Vert _E\\\le & {} \Vert \mathbf{A}^{\frac{\mathrm{T}}{2}}(\mathbf{G}+\nu \mathbf{M})^{-1}{} \mathbf{U}^{\frac{1}{2}}\Vert _E\Vert \mathbf{U}^{\frac{\mathrm{T}}{2}}\hat{\mathfrak {a}}\Vert _E\\= & {} \hat{\eta }\left\| \hat{\alpha }_h^k\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}\\\le & {} \hat{\eta }\left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}, \end{aligned}$$

which enables us the $V^1L^2$ estimates for the scheme (10).

For $\nu =1$, $\nu =0.1$ and $\nu =0.01$ in $\varOmega =(0,1)$ and $J=(0,1)$, Tables 1 and 2 show verified results of $\eta$ and $\hat{\eta }$, respectively. We take the basis of finite element subspaces $S_h(\varOmega )$ and $V_k^1(J)$ are taken as P1-function with uniform mesh on $\varOmega$ and J, respectively. By the verified computing results, we can conclude that the projection $\hat{Q}_h^k$ is not $V^1$-stable, and our proposed projection $Q_h^k$ satisfies $V^1$-stability as well as it has $L^2H^1_0$-stability.

Table 1 The numerical results $\eta$ in $\varOmega =(0,1)$, $J=(0,1)$

Table 2 The numerical results $\hat{\eta }$ in $\varOmega =(0,1)$, $J=(0,1)$

4 Constructive error estimates

In this section, we consider constructive error estimates of the projection $Q_h^k$ for the finite element approximation. For an arbitrary $u \in V(\varOmega ,J) \cap L^2\bigl (J;X(\varOmega )\bigr )$, we define the projection $\bar{P}_h^ku\in S_h^k(\varOmega ,J)$ satisfying the following weak form:

$$\begin{aligned}&\left<{\frac{\partial {}}{\partial {t}}}\bar{P}_h^ku,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}+\nu \left<\nabla P_h u,{\frac{\partial {}}{\partial {t}}} \nabla v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d}\nonumber \\&\quad =\left<\varDelta _tu,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}, \end{aligned}$$

(12)

for all $v_h^k\in S_h^k(\varOmega ,J)$. Note that, for a fixed $v_h^k\in S_h^k(\varOmega ,J)$, by taking $v_h$ as $v_h={\frac{\partial {}}{\partial {t}}}v_h^k$ in (4), we have

$$\begin{aligned}&\left<{\frac{\partial {}}{\partial {t}}}P_hu,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )} + \nu \left<\nabla P_hu,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d}\nonumber \\&\quad = \left<\varDelta _tu,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}. \end{aligned}$$

(13)

From (12) and (13), we obtain

$$\begin{aligned} \left<{\frac{\partial {}}{\partial {t}}}P_hu,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}= \left<{\frac{\partial {}}{\partial {t}}}\bar{P}_h^ku,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}. \end{aligned}$$

(14)

Moreover, from the definition of $V^1$-projection, we have

$$\begin{aligned} \left<{\frac{\partial {}}{\partial {t}}}P_hu,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}= \left<{\frac{\partial {}}{\partial {t}}}P_1^kP_hu,{\frac{\partial {}}{\partial {t}}}v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )}. \end{aligned}$$

(15)

From (14) and (15), it follows that $\bar{P}_h^k=P_1^kP_h$ because $P_1^kP_hu\in S_h^k(\varOmega ,J)$.

Remark 1

[1, Lemma 2.2] If $V_k^1(J)$ is the P1-finite element space, then $P^k_1$ coincides with $\varPi _k$, it follows that $\bar{P}_h^k=P_h^k(=\varPi _kP_h)$.

For the projection $Q_h^k$, from the triangle inequality, we have

$$\begin{aligned} \left\| u-Q_h^ku\right\| _{L^2\bigl (J;H_0^1(\varOmega )\bigr )}\le & {} \left\| u-P_h^ku\right\| _{L^2\bigl (J;H_0^1(\varOmega )\bigr )}+\left\| P_h^ku-Q_h^ku\right\| _{L^2\bigl (J;H_0^1(\varOmega )\bigr )},\\ \left\| u-Q_h^ku\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}\le & {} \left\| u-P_h^ku\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}+\left\| P_h^ku-Q_h^ku\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )},\\ \left\| u(T)-Q_h^ku(T)\right\| _{L^2(\varOmega )}\le & {} \left\| u(T)-P_hu(T)\right\| _{L^2(\varOmega )}+\left\| P_h^ku(T)-Q_h^ku(T)\right\| _{L^2(\varOmega )}, \end{aligned}$$

where we have used the fact that $P_h^ku(T)=\varPi _kP_hu(T)=P_hu(T)$. Thus we now present the estimation for $P_h^ku-Q_h^ku\in S_h^k(\varOmega ,J)$ below.

From (8) and (12), and letting $\delta _h^k:=\bar{P}_h^ku-Q_h^ku\in S_h^k(\varOmega ,J)$, we can obtain

$$\begin{aligned} a_0(\delta _h^k,v_h^k) =\nu \left<\nabla \xi ,{\frac{\partial {}}{\partial {t}}}\nabla v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d} \end{aligned}$$

(16)

where $\xi :=\bar{P}_h^ku-P_h u\in V$. Moreover, we define $\beta _h^k\in S_h^k(\varOmega ,J)$ satisfying

$$\begin{aligned}&\left<{\frac{\partial {}}{\partial {t}}}\nabla \beta _h^k,{\frac{\partial {}}{\partial {t}}}\nabla v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d}=\left<\nabla \xi ,{\frac{\partial {}}{\partial {t}}}\nabla v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d},\\&\quad \forall v_h^k \in S_h^k(\varOmega ,J). \end{aligned}$$

Note that by taking $v_h^k=\beta _h^k$ on the above, it follows that $\left\| {\frac{\partial {}}{\partial {t}}}\beta _h^k\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}\le \left\| \xi \right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}$. Then we obtain

$$\begin{aligned} a_0(\delta _h^k,v_h^k) =\nu \left<{\frac{\partial {}}{\partial {t}}}\nabla \beta _h^k,{\frac{\partial {}}{\partial {t}}}\nabla v_h^k\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d} \end{aligned}$$

(17)

We now define the matrices $\mathbf{W}$ and $\mathbf{Y}$ in $\mathbb {R}^{mn \times mn}$ by

$$\begin{aligned} \mathbf{W}_{i,j}:= & {} \left<{\frac{\partial {}}{\partial {t}}}\nabla \varphi _j,{\frac{\partial {}}{\partial {t}}} \nabla \varphi _i\right>_{L^2\bigl (J;L^2(\varOmega )\bigr )^d},\quad \mathbf{Y}_{i,j}:=\left<\varphi _j(\cdot ,T),\varphi _i(\cdot ,T)\right>_{L^2(\varOmega )},\\&\forall i,j \in \{1,\ldots ,mn\}, \end{aligned}$$

respectively. Since the matrix $\mathbf{W}$ is symmetric and positive definite, we can denote the Cholesky decomposition as $\mathbf{W}=\mathbf{W}^{\frac{1}{2}}{} \mathbf{W}^{\frac{\mathrm{T}}{2}}$. Moreover, since the matrix $\mathbf{Y}$ is symmetric and positive semi-definite, we can decompose it as $\mathbf{Y}=\mathbf{Y}^{\frac{1}{2}}{} \mathbf{Y}^{\frac{\mathrm{T}}{2}}$. From the fact that $\delta _h^k$ and $\beta _h^k$ in $S_h^k(\varOmega ,J)$, there exist coefficient vectors $\mathfrak {d}:=(\mathfrak {d}_1,\ldots ,\mathfrak {d}_{mn})^{\mathrm{T}}$ and $\mathfrak {b}:=(\mathfrak {b}_1,\ldots ,\mathfrak {b}_{mn})^{\mathrm{T}}$ in $\mathbb {R}^{mn}$ such that $\delta _h^k= \sum _{i=1}^{mn} \mathfrak {d}_i\varphi _i = \varphi ^{\mathrm{T}}\mathfrak {d}$ and $\beta _h^k= \sum _{i=1}^{mn} \mathfrak {b}_i\varphi _i = \varphi ^{\mathrm{T}}\mathfrak {b}$. Then, the variational equation (17) is equivalent to the following.

$$\begin{aligned} (\mathbf{A}+\nu \mathbf{B})\mathfrak {d}=\nu \mathbf{W}\mathfrak {b}. \end{aligned}$$

(18)

Let

$$\begin{aligned} \gamma _1:= & {} \nu \Vert \mathbf{M}^{\frac{\mathrm{T}}{2}}(\mathbf{A}+\nu \mathbf{B})^{-1}{} \mathbf{W}^{\frac{1}{2}}\Vert _E,\\ \gamma _0:= & {} \nu \Vert \mathbf{U}^{\frac{\mathrm{T}}{2}}(\mathbf{A}+\nu \mathbf{B})^{-1}{} \mathbf{W}^{\frac{1}{2}}\Vert _E,\\ \gamma _T:= & {} \nu \Vert \mathbf{Y}^{\frac{\mathrm{T}}{2}}(\mathbf{A}+\nu \mathbf{B})^{-1}{} \mathbf{W}^{\frac{1}{2}}\Vert _E. \end{aligned}$$

Then we have the following main result in this paper.

Theorem 2

Assume that $V_k^1(J)$ is the P1 finite element space. Let $u\in V(\varOmega ,J) \cap L^2\bigl (J;X(\varOmega )\bigr )$ be a solution of (1) for $f\in L^2\bigl (J;L^2(\varOmega )\bigr )$. Then, we have the following estimations.

$$\begin{aligned} \left\| u-Q_h^ku\right\| _{L^2\bigl (J;H_0^1(\varOmega )\bigr )}\le & {} \tilde{C}_1(h,k)\left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )},\\ \left\| u-Q_h^ku\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}\le & {} \tilde{C}_0(h,k)\left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )},\\ \left\| u(T)-Q_h^ku(T)\right\| _{L^2(\varOmega )}\le & {} \tilde{c}_0(h)\left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}, \end{aligned}$$

where

$$\begin{aligned} \tilde{C}_1(h,k)\equiv & {} C_1(h,k)+C_J(k)C_{inv}(h)\gamma _1,\\ \tilde{C}_0(h,k)\equiv & {} C_0(h,k)+C_J(k)C_{inv}(h)\gamma _0,\\ \tilde{c}_0(h,k)\equiv & {} c_0(h)+C_J(k)C_{inv}(h)\gamma _T. \end{aligned}$$

Proof

From (18), we can obtain

$$\begin{aligned} \left\| \delta _h^k\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}= & {} \left\| \mathbf{M}^{\frac{\mathrm{T}}{2}}\mathfrak {d}\right\| _E =\nu \left\| \mathbf{M}^{\frac{\mathrm{T}}{2}}(\mathbf{A}+\nu \mathbf{B})^{-1}{} \mathbf{W}\mathfrak {b}\right\| _E \le \gamma _1\left\| \mathbf{W}^{\frac{\mathrm{T}}{2}}\mathfrak {b}\right\| _E\\ \left\| \delta _h^k\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}= & {} \left\| \mathbf{U}^{\frac{\mathrm{T}}{2}}\mathfrak {d}\right\| _E =\nu \left\| \mathbf{U}^{\frac{\mathrm{T}}{2}}(\mathbf{A}+\nu \mathbf{B})^{-1}{} \mathbf{W}\mathfrak {b}\right\| _E \le \gamma _0\left\| \mathbf{W}^{\frac{\mathrm{T}}{2}}\mathfrak {b}\right\| _E,\\ \left\| \delta _h^k(T)\right\| _{L^2(\varOmega )}= & {} \left\| \mathbf{Y}^{\frac{\mathrm{T}}{2}}\mathfrak {d}\right\| _E =\nu \left\| \mathbf{Y}^{\frac{\mathrm{T}}{2}}(\mathbf{A}+\nu \mathbf{B})^{-1}{} \mathbf{W}\mathfrak {b}\right\| _E \le \gamma _T\left\| \mathbf{W}^{\frac{\mathrm{T}}{2}}\mathfrak {b}\right\| _E. \end{aligned}$$

Moreover, we have

$$\begin{aligned} \Vert \mathbf{W}^{\frac{{\mathrm{T}}}{2}}\mathfrak {b}\Vert _E=\left\| {\frac{\partial {}}{\partial {t}}}\beta _h^k\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}\le \left\| \xi \right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}. \end{aligned}$$

Note that $\bar{P}_h^k=P_h^k(=\varPi _kP_h)$ from Remark 1. Then it follows that

$$\begin{aligned} \left\| \xi \right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}= & {} \left\| \bar{P}_h^ku-P_h u\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}\\= & {} \left\| \varPi _kP_hu-P_h u\right\| _{L^2\bigl (J;H^1_0(\varOmega )\bigr )}\\\le & {} C_{inv}(h)\left\| \varPi _kP_hu-P_h u\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}\\\le & {} C_J(k)C_{inv}(h)\left\| P_hu\right\| _{V^1\bigl (J;L^2(\varOmega )\bigr )}\\\le & {} C_J(k)C_{inv}(h)\left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}, \end{aligned}$$

where we have used the fact that $\left\| P_hu\right\| _{V^1\bigl (J;L^2(\varOmega )\bigr )}\le \left\| f\right\| _{L^2\bigl (J;L^2(\varOmega )\bigr )}$ in [3]. Therefore, the proof is completed from Theorem 1 and the fact $\delta _h^k=P_h^ku-Q_h^ku$. $\square$

5 Numerical results

In this section, we show some numerical results of $\gamma _1$, $\gamma _0$ and $\gamma _T$ in Theorem 2. We take the basis of finite element subspaces $S_h(\varOmega )$ and $V_k^1(J)$ are taken as P1-function with uniform mesh on $\varOmega$ and J, respectively. Tables 3, 4 and 5 show verified results of $\gamma _1$, $\gamma _0$ and $\gamma _T$ for $\nu =1$, $\nu =0.1$ and $\nu =0.01$ in $\varOmega =(0,1)$ and $J=(0,1)$. From the verified results in Tables 3, 4 and 5, we may conclude that $\gamma _0$ and $\gamma _T$ are dependent on the parameter $\nu$ more clearly than $\gamma _1$, but asymptotically converge to some fixed constants when h and k tend to zero.

Table 3 The numerical results in $\varOmega =(0,1)$, $J=(0,1)$

Table 4 The numerical results in $\varOmega =(0,1)$, $J=(0,1)$

Table 5 The numerical results in $\varOmega =(0,1)$, $J=(0,1)$

All computations in Tables are carried out on the Dell Precision 5820 Intel Xeon CPU 4.0GHz by using INTLAB(Var. 10.1), a tool box in MATLAB(Var. R2017b) developed by Rump [6] for self-validating algorithms. Therefore, all numerical values in these tables are verified data in the sense of strictly rounding error control.

6 Conclusion

We presented a new full-discrete finite element projection $Q_h^k$ for the heat equation, and derived the constructive stability by the numerical computations with guaranteed accuracy. Our scheme is closely related to that in [5] in the sense that the constructive error estimates is established by using the results obtained by the same paper. Therefore, it is considered as an another version of [5], but the present scheme should be more familiar method to researchers working on numerical analysis. Namely, it is not necessary to have any complicated manipulation for the verified computation of matrix function.

Particularly, the estimate $\left\| u(T)-Q_h^ku(T)\right\| _{L^2(\varOmega )}$ should be useful to the verified computation for nonlinear problems by the time-evolutional method (cf. [2]), which will be presented in our forthcoming paper. Thus, our method will play an important role in the numerical verification method to find exact solutions for the nonlinear parabolic equations.