Abstract
In this paper, we discuss a hybridised method for FEM-BEM coupling. The coupling from both sides use a Nitsche-type approach to couple to the trace variable. This leads to a formulation that is robust and flexible with respect to approximation spaces and can easily be combined as a building block with other hybridised methods. Energy error norm estimates and the convergence of Jacobi iterations are proved and the performance of the method is illustrated on some computational examples.
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1 Introduction
The coupling of finite element (FEM) and boundary element (BEM) methods is the most widely used approach for solving multi-physical problems on an unbounded domain. It allows to take advantage of both methods. On the one hand, the BEM reduces the dimension of the problem by using the boundary integral equation, hence it is commonly used in exterior unbounded domains. On the other hand, the FEM is known for its robustness and universal applicability even for problem of inhomogeneous or non-linear nature.
The first coupled procedure was introduced by Zienkiewicz, Kelly and Bettess [38]. It was analysed by Brezzi, Johnson and Nédélec [5], [4] and [24] for problem in unbounded domains. It is often referred to as the Johnson-Nédélec coupling. Extension for higher order equations was considered by Wendland [37]. The convergence analysis requires compactness of the double layer potential that can be obtained on smooth boundaries. Furthermore, even for a symmetric discretisation scheme, the coupling method produces a system of equations with a non-symmetric coefficient matrix.
In order to avoid these disadvantages, a symmetric coupling of FEM and BEM was devised by Costabel [14] and Han [22]. The independence of the compactness condition was obtained by using both equations of the Calderón system, contrary to the previously introduced methods that employ only one of the two equations of the Calderón system. Some years later, Sayas [32] showed that the weaker assumption of a Lipschitz coupling interface is sufficient for the Johnson-Nédélec coupling. His analysis has since been simplified by Steinbach [35].
More recent developments have focused on the coupling of BEM with mixed FEM. In [6] and [27] the authors analysed symmetric coupling of BEM and mixed FEM that uses Raviart-Thomas elements. Further work on coupling BEM with mixed FEM with such elements as Brezzi-Douglas-Marini or Brezzi-Douglas-Marini-Fortin was proposed by Carstensen and Funken [9].
Gatica, Heuer and Sayas [21] and [20] introduced the first coupling of BEM and discontinuous Galerkin (DG) methods, in order to exploit the possibility to easily use high order approximation in the latter. Another coupling of interior penalty DG methods with BEM was presented by Of, Rodin, Steinbach and Taus [29]. A general approach using the unified hybridisation technique was presented by Cockburn and Sayas [13]. The class of FEM considered includes the mixed, the DG and the hybridisable discontinuous Galerkin (HDG) methods. Further collaboration of these authors with Gúzman led to a new convergence result published in [12].
In this paper, we present the coupling of FEM and BEM using weak imposition of coupling conditions. Nitsche’s method [28] is widely used in the context of FEM for imposing boundary conditions. In addition, methods based on Nitsche’s approach have been successfully utilised for BEM domain decomposition problems in [19] and [10], and more recently for weakly imposing boundary conditions for BEM in [2]. Merging these two approaches for FEM and BEM we weakly impose both coupling conditions in a hybridised formulation on the boundary. The hybridisation is made by introducing a trace variable and imposing the coupling in the form of a Nitsche-type Dirichlet condition on the two systems. The use of Nitsche’s method allows us to use the Dirichlet trace as the hybrid variable, ensuring continuity by a consistent penalty term. The test function partner of the trace variable then acts to ensure continuity of fluxes. The global system can be constructed using arbitrary approximation orders for the two sub systems and the trace variable and the sub problem can be solved independently. The stability of the method poses no constraint on the approximation spaces and mesh refinement does not require special treatment as in the case of Johnson-Nédélec coupling. This means that the two systems can have independent meshes, that both must be integrated only against the trace variable. We here consider the standard continuous FEM, but the formulation is by and large agnostic to the choice of FEM used in the bulk and the method can be applied with discontinuous FEM as well, such as DG, HDG [11], or HHO [16] using a hybridised coupling on the interior domain boundary. In the case of using discontinuous FEM, our formulation can be interpreted as a hybridised interior penalty formulation of the class of methods discussed in [13]. Finally, we note that, thanks to the use of Nitsche-type mortaring, the method proposed herein can be used in the framework for unfitted hybridised methods introduced in [7]. In that case a surface mesh is required for the definition of the BEM method, but the FEM approximation on the interior domain can be computed on an unfitted bulk discretisation.
As many existing approaches of coupling FEM and BEM, we use Finite Element Tearing and Interconnecting (FETI) and Boundary Element Tearing and Interconnecting (BETI) type of methods [25] to solve the reduced system for the hybrid variable. FETI is formulated using a Schur complement formulation, while BETI is usually formulated in terms of Steklov-Poincaré operators. Although Nitsche’s method is an established framework for domain decomposition for finite elements methods such as FETI, it was not recognised by BETI community. In this paper, we demonstrate how the hybrid Nitsche approach can be integrated into the BETI framework.
The rest of the paper is organised as follows. We introduce the model problem in this section. In Section 2, we present on continuous level the symmetric coupling of BEM and FEM formulation known from [14] and [22]. The discrete formulation including weakly imposed coupling condition is introduced and analysed in Section 3. Although the formulation obtained is not symmetric, we comment of how symmetry can be obtained for the associated Steklov-Poincaré operator. We discuss iterative domain decomposition in the model case of a simple relaxed Jacobi algorithm in Section 4 and prove its convergence. In Section 5, we present some numerical results, and we conclude with some remarks in Section 6.
1.1 Model problem
Let us consider the unbounded domain \({\Omega } = \mathbb {R}^{3}\). We divide Ω into a bounded internal part Ω− and an unbounded external part Ω+ with common Lipschitz boundary Γ, with n the outer unit normal vector of the domain Ω− on Γ. We let \(\partial _{n} u := \tfrac {\partial u}{\partial _{n}}\) denote the outward normal derivative, f ∈ L2(Ω) be a function with support in Ω− and introduce a function \(\varepsilon \in L^{\infty }({\Omega })\), 𝜖 ≥ 0. Then, we can formulate our model problem as follows
where \(u^{i} = u|_{{\Omega }^{i}}\), i ∈{−,+}. The function ε is introduced to make the interior problem heterogeneous and hence unsuitable for treatment using the boundary element method alone.
Remark 1
It is straightforward to extend the discussion to the case with a smoothly varying diffusion coefficient in Ω− which has a jump over Γ.
Let us begin by introducing the variational formulation of the coupled system.
2 Variational formulation
Let \(\left <\cdot ,\cdot \right >_{\Gamma }\) denote the L2(Γ)-inner product that can be extended to a duality pairing on \(H^{-\frac {1}{2}}({\Gamma }) \times H^{\frac {1}{2}}({\Gamma })\).
We start with the variational formulation of the internal problem. Applying integration by parts for first equation of (1) for every \(v \in {H_{0}^{1}}({\Omega }^{-})\) we have
We define the Green’s function for the Laplace operator in \(\mathbb {R}^{3}\) as follows
In this paper, we focus on the problem in \(\mathbb {R}^{3}\). A similar analysis can be used for problems in \(\mathbb {R}^{2}\), in which case this definition should be replaced by \(G(x,y) := \frac {\log {| x - y |}}{2\pi }\). Following the standard approach (see, e.g., [34, Chapter 6]), we introduce single layer and double layer operators \(\mathcal {V}:H^{-\frac {1}{2}}({\Gamma }) \rightarrow H^{1}({\Omega }^{+})\) and \(\mathcal {K}:H^{\frac {1}{2}}({\Gamma }) \rightarrow H^{1}({\Omega }^{+})\) respectively as
where x ∈Ω+ ∖Γ and ny is an outer unit normal vector (for Ωi, i ∈{−,+}) in the point y.
Following [34, Chapter 1], we define the Dirichlet and Neumann traces
where \(H^{1}({\Delta },{\Omega }^{i}) := \left \{v \in H^{1}({\Omega }^{i}): {\Delta } v \in L^{2}({\Omega }^{i})\right \}\), for i ∈{−,+}, and nx is an outer (for Ω−) normal vector to Γ in the point x. The following results will be useful in what follows.
Lemma 1 (Trace theorem)
Let i ∈{−,+}, then for \({\Omega }^{i} \subset \mathbb {R}^{3}\) the trace operator \({\gamma _{D}^{i}}: H^{1}({\Omega }^{i}) \rightarrow H^{\frac {1}{2}}({\Gamma })\) is bounded for all v ∈ H1(Ωi)
Proof
See [26, Theorem 3.37]. □
We use {⋅}Γ to denote an average of the interior and exterior traces of a function. Then, applying the trace mappings yields to the single-layer, double-layer, and adjoint double-layer potentials and hypersingular boundary integral operator
For the solution u of the problem (1), we have the following boundary integral equations on Γ
where \(C^{\pm }: H^{\frac {1}{2}}({\Gamma }) \times H^{-\frac {1}{2}}({\Gamma }) \rightarrow H^{\frac {1}{2}}({\Gamma }) \times H^{-\frac {1}{2}}({\Gamma })\) denotes two Calderón projectors defined as follows
From the relation (4) for external traces we can construct the following exterior Dirichlet-to-Neumann operator
Obviously, it makes sense only if the inverse of the operator V exists.
Using Dirichlet-to-Neumann operator (5) we introduce a new variable \(\lambda = \gamma _{N}^{+}u = \partial _{n} u^{+}\) as
The classical symmetric coupling that satisfies the transmission conditions of (1) is as follows
Find u ∈ H1(Ω−) and \(\lambda \in H^{-\frac {1}{2}}({\Gamma })\) such that for all v ∈ H1(Ω−) and \(\zeta \in H^{-\frac {1}{2}}({\Gamma })\)
For simplicity, we omit Dirichlet trace for integration over boundary.
2.1 Well-posedness of the continuous problem
The following results are well known (see [14] or [22]), but for reader’s convenience we present them in the case of problem (1). Let us propose a more compact formulation of (7).
Find u ∈ H1(Ω−) and \(\lambda \in H^{-\frac {1}{2}}({\Gamma })\) such that for all v ∈ H1(Ω−) and \(\zeta \in H^{-\frac {1}{2}}({\Gamma })\)
where
For simplicity we introduce the space \(\mathbb {V} := H^{1}({\Omega }^{-}) \times H^{-\frac 12}({\Gamma })\) and the associated norm
We begin by showing the boundedness of the bilinear form A.
Lemma 2 (Boundedness)
There exists constant β > 0 such that for all w,v ∈ H1(Ω−) and \(\lambda , \varphi \in H^{-\frac 12}({\Gamma })\)
Proof
We use the Cauchy-Schwarz inequality, the duality pairing relation and continuity of boundary operators (see [34, Section 6.2–6.5]) to obtain
We finished by applying the trace inequality (3) to terms including \(\left \|\cdot \right \|_{H^{\frac {1}{2}}({\Gamma })}\) norm,
□
The next step in proving the existence and uniqueness of the solution is coercivity of the bilinear form A.
Lemma 3 (Coercivity)
There exists constant α > 0 such that for all v ∈ H1(Ω−) and \(\varphi \in H^{-\frac 12}({\Gamma })\)
Proof
Consider the form \(A\left (\left (v, \varphi \right ), \left (v, \varphi \right )\right )\). Because of the relation between the double layer potential and its adjoint, the associated terms cancel. Using coercivity of V (see [34, Theorem 6.22]), and coercivity of W on functions with zero average (see [34, Theorem 6.24] we obtain
where \(\bar v = |{\Gamma }|^{-1} {\int \limits }_{\Gamma } v ~\mathrm {d} s\) and \(c_{\varepsilon } = \min \limits (1,\varepsilon )\). This shows (11) when ε > 0. For the case 𝜖 = 0 we need a Poincaré inequality of the form
where g(v) is some functional that is non-zero for constant non-zero v [18, Lemma B.63]. We claim that this holds with \(g(v) = \left < V \varphi (v), \varphi (v) \right >_{\Gamma }\) where φ(v) is defined by the second equation of (7). We immediately see that if this is true then there exists α > 0 such that
We need to show that for constant v≠ 0 there holds \(\left < V \varphi (v), \varphi (v) \right >_{\Gamma } \ne 0\). Let v|Γ = u0 be a non-zero constant. Then, we need to study
We argue by contradiction. Assume that φ(v) = 0. Then, u0 is the trace of a solution to the homogeneous Neumann problem in Ω−. However by the first line of the left relation of (4) there holds for all such traces
Hence the functional is defined by
However since the operator on the left-hand side, defined by, V is injective it follows that φ(v)≠ 0, which leads to a contradiction. Hence \(\left < V \varphi (v), \varphi (v) \right >_{\Gamma } \ne 0\) for v constant. This concludes the proof. □
The existence and uniqueness of the solution of problem (8) is achieved by using the Lax-Milgram theorem.
3 The discrete problem
In the previous section, we focused on the classical symmetric formulation. Now, using this formulation we propose the weak penalty formulation of a couple problem. The discretisation is made by using finite element methods inside the domain and boundary integral methods outside the domain. We prove the well-posedness of the discrete problem and analyse the convergence of the proposed method.
We assume that Ω− is a polyhedral domain. The boundary Γ may be decomposed in a set of n planar surfaces \(\{{\Gamma }_{j}\}_{j=1}^{n}\). It will be convenient to use following broken Sobolev spaces over the polyhedral boundary Γ. For s > 1 define
When s > 1 the norm on \(\widetilde H^{s}({\Gamma })\), is defined as the broken norm over the faces of the polyhedral boundary Γ
When 0 ≤ s ≤ 1 the space \(\widetilde H^{s}({\Gamma })\) coincides with the usual space Hs(Γ) and their norms are the same (for more details see [31, Definition 4.1.48]).
Let \(\mathcal {T}_{h}\) be a triangulation of \(\overline {\Omega }^{-}\) made of tetrahedrons. For each of element \(K \in \mathcal {T}_{h}\), hK := diam(K), and \(h := \max \limits _{K \in \mathcal {T}_{h}} h_{K}\). Let \(\mathcal {G}_{i}\), i = 1,2 denote two different surface triangulation of the boundary Γ. For notation convenience we here assume that the trace mesh of \(\mathcal {T}_{h}\) and the \(\mathcal {G}_{i}\) all have similar local mesh size.
The following result will be useful in what follows.
Lemma 4 (Trace inequality)
There exists Cmax > 0, independent of hK, such that for all \(K \in \mathcal {T}_{h}\) and polynomial function v in K the following discrete trace inequality holds
Proof
See [15, Lemma 1.46]. □
To discretise the problem (7) over the triangulation one can choose either continuous or discontinuous finite elements. For simplicity of the analysis we choose the following spaces
Let us denote \(\mathcal {V}_{h} := {V_{h}^{j}} \times {W_{h}^{k}}\) and \(v_{h} = \left (v_{h}^{-}, v_{h}^{+}\right ) \in \mathcal {V}_{h}\). Using the above spaces we propose the hybrid discrete formulation of the problem (7)
Find \(\left (u_{h}, \lambda _{h}, \widetilde {u}_{h}\right ) \in \mathcal {V}_{h}\times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\) such that for all \(\left (v_{h}, \varphi _{h}, \widetilde {v}_{h}\right ) \in \mathcal {V}_{h} \times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\)
where
The formulation of bilinear form ah is well known for example from [17]. As usual for symmetric Nitsche methods the stabilisation parameter τF > 0 has to be chosen sufficiently large for stability.
Remark 2 (Impedance boundary condition)
The hybrid weakly imposed Dirichlet and Neumann boundary conditions is related to an impedance boundary condition of the type
with \(\gamma = \frac {h}{\tau _{F}} = {\tau _{B}}\). This can be seen considering terms associated with \(\widetilde {v}_{h}\) in above definition of the bilinear forms.
Remark 3 (Relation to a standard Nitsche-type method without hybridisation)
To transform the hybridised method to a Nitsche-type method without hybridisation we proceed as follows. The trace variable \(\widetilde u_{h}\) is eliminated by replacing it by a linear combination of \(u_{h}^{+}\) and \(u_{h}^{-}\) (i.e. replace \(\widetilde u_{h}\) with \(\omega u_{h}^{+} + (1- \omega ) u_{h}^{-}\), ω ∈ [0,1]) and similarly replace the test function \(\widetilde v_{h}\) replaced by the same linear combination of the test functions \(v_{h}^{+}\) and \(v_{h}^{-}\) (i.e. replace \(\widetilde v_{h}\) with \(\omega v_{h}^{+} + (1- \omega ) v_{h}^{-}\)) (see [8, Section 4.2]). The below analysis carries over verbatim to this case.
3.1 Symmetric formulation
Despite using the symmetric Nitsche method, our whole system is not symmetric. This is a consequence of the lack of symmetry of the boundary element method with weak imposition. We can use the Steklov-Poincaré operator to eliminate the flux variable, so that the non-symmetric method above is transformed into a symmetric reduced system as we show below.
The following equations are associated with bilinear form bh from (15) reads
Similar to the continuous formulation we use the Dirichlet-to-Neumann operator (5) to obtain
Injecting this relation into the second equation leads to the formulation of the new symmetric bilinear form
For clarity we carry out the analysis of the discrete problem using non-symmetric formulation.
3.2 Well-posedness of the discrete problem
Let us consider the following norms
Lemma 5 (Equivalence of the norms)
For all \(\left ({w_{h}}, \lambda _{h}, {\widetilde {w}_{h}}\right ) \in \mathcal {V}_{h} \times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\) there exist positive constants \(C_{\mathcal {F}}, C_{{\mathscr{B}}}\) such that
Proof
For (19) we use the trace inequality (14) and for (20) we use the inverse inequality \( h^{\frac 12} \left \| \lambda _{h}\right \|_{L^{2}({\Gamma })} \leq C \left \|\lambda _{h}\right \|_{H^{-\frac 12}({\Gamma })}\) (see [31, Remark 4.4.4]). □
After introducing the norms that we use we can start by showing the boundedness of the bilinear forms ah and bh.
Lemma 6 (Boundedness)
There exists positive constant \(\beta _{\mathcal {F}}\) such that for all \(w,v \in H^{\frac 32+\delta }({\Omega }^{-})\), for δ > 0, and \(\widetilde {w}, \widetilde {v} \in L^{2}({\Gamma })\)
There exists positive constant \(\beta _{{\mathscr{B}}}\) such that for all \(w,v \in H^{\frac {1}{2}}({\Gamma })\), λ,φ ∈ L2(Γ) and \(\widetilde {w}, \widetilde {v} \in L^{2}({\Gamma })\)
Proof
We use Cauchy-Schwarz inequality to obtain (21). In the case of (22), we use Cauchy-Schwarz inequality and Lemma 2. □
To show the discrete well-posedness of (15) we need the ellipticity of the bilinear forms ah and bh. Our formulation contains two stabilisation parameters τF and τB. The first parameter associated with finite element bilinear form must be chosen to allow us to use trace inequality (14). On the other hand, for τB for the bilinear form bh positivity is the only constraint.
Lemma 7 (Coercivity)
Assume that τB > 0 and positive constant τF is large enough. Then, there exists positive constant α such that for all \(\left ({w_{h}}, \lambda _{h}, {\widetilde {w}_{h}}\right ) \in \mathcal {V}_{h} \times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\)
Proof
Let us start with bilinear form ah. First we assume that ε ≥ εmin > 0
Using Cauchy-Schwarz and trace inequality (14), followed by Young’s inequality, we arrive at
We finish by applying the equivalence of the norms (19) under the assumption that \({\tau _{F}} > 2 C_{max}^{2}\).
In the case of the bilinear form bh, by using the results from Lemma 3 we obtain, with \(\bar w_{h} = |{\Gamma }|^{-1} {\int \limits }_{\Gamma } w^{+}_{h} ~\mathrm {d}s\),
Observe that when ε > 0 we may bound
where we applied the trace inequality (3) in the last step. The right-hand side is controlled by the lower bounds on ah and bh above. Once again, we finish by applying the equivalence of the norms (20).
In case ε = 0 we need to show that a Poincaré inequality holds, similar to (12), this time on the form
To this end, since coercivity holds up to a constant, we may assume that \(w^{-}_{h}\vert _{\Gamma } = \tilde w_{h} = w^{+}_{h} = \bar w_{h}\) and proceed verbatim as in the continuous case, since in that case the continuous and discrete expressions corresponding to (13) are the same. □
The existence and uniqueness of the solution of problem (15) is achieved by using the Lax-Milgram theorem. In addition, the proposed method is consistent for sufficiently smooth exact solutions as the following result shows.
Lemma 8 (Consistency)
Let δ > 0, \(u^{-} \in H^{\frac 32+\delta }({\Omega }^{-})\), \(u^{+} \in H^{\frac {1}{2}}({\Gamma })\) and ∂nu+ = λ ∈ L2(Γ) be the solution of problem (1) and \(\widetilde {u} = u^{-} = u^{+}\) on Γ. If \(\left (u_{h}, \lambda _{h}, \widetilde {u}_{h}\right ) \in \mathcal {V}_{h}\times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\) solves (15) then, for all \(\left (v_{h}, \varphi _{h}, \widetilde {v}_{h}\right ) \in \mathcal {V}_{h} \times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\) the following holds
Proof
Because of the transmission conditions from (1) we have u = u+ = u− and ∂nu = ∂nu+ = ∂nu− on Γ
By adding above expressions and using the facts that λ = ∂nu and \(\widetilde {u} = u\) on Γ, we obtain consistency. □
3.3 Error analysis
In this section we present the error estimates for the method. These estimates are proved using the following norm
The first step is the following version of Cea’s lemma.
Proposition 1 (Cea’s Lemma)
Let δ > 0, \(u^{-} \in H^{\frac 32+\delta }({\Omega }^{-})\), \(u^{+} \in H^{\frac {1}{2}}({\Gamma })\) and ∂nu+ = λ ∈ L2(Γ) be the solution of problem (1), \(\widetilde {u} = u^{-} = u^{+}\) on Γ, and let \(\left (u_{h}, \lambda _{h}, \widetilde {u}_{h}\right ) \in \mathcal {V}_{h}\times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\) solve (15). Then, there exists C > 0, independent of h, such that
Proof
Let us denote
By Lemma 7 there exists α > 0, independent of h, such that for all \(\left (v_{h}, \varphi _{h}, \widetilde {v}_{h}\right ) \in \mathcal {V}_{h} \times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\) there exists \(\left (w_{h}, \phi _{h}, \widetilde {w}_{h}\right ) \in \mathcal {V}_{h}\times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\) with \(\left \|\left (w_{h}, \phi _{h}, \widetilde {w}_{h}\right )\right \|_{h}=1\), and
Now using Lemma 6, we get continuity of Ah, there exists β > 0
Let \(\left (v_{h}, \varphi _{h}, \widetilde {v}_{h}\right ) \in \mathcal {V}_{h} \times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\). Then, using the triangle inequality we see that
and it follows from (27), Lemma 8 and (28) that
Thus, we get (26) with \(C := 1+\tfrac {\beta }{\alpha }\). □
Lemma 9 (Energy norm estimates)
For \(s > \tfrac 32\), r > 1 and \(p \geq \tfrac 12\), let u−∈ Hs(Ω−), \(u^{+} \in \widetilde H^{r}({\Gamma })\) and \(\partial _{n} u^{+} = \lambda \in \widetilde H^{p}({\Gamma })\) be the solution of problem (1). On Γ there holds \(u^{-} = u^{+} = \widetilde {u} \) on Γ. Let \(\left (u_{h}, \lambda _{h}, \widetilde {u}_{h}\right ) \in \mathcal {V}_{h}\times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\) solve (15). If the mesh is quasi uniform, then there exists C > 0, independent of h, such that
where \(\sigma = {\min \limits } \{j+1, s\}\), \(\phi = \min \limits \{k+1, r\}\), \(\xi = {\min \limits } \{m+1, r\}\) and \(\psi = {\min \limits } \{l+1, p\}\).
Proof
The result is a consequence of (26) and approximation. Applying triangle inequality and trace inequality [3, Theorem 1.6.6] followed by Young’s inequality, we obtain
Using approximation results [18, Theorem 1.109] and [31, Theorem 4.3.19] for the second term containing hybrid variables, we claim
For the boundary part, by applying triangle inequality, we obtain
Using approximation results [31, Theorems 4.3.19, 4.3.20 and 4.3.22] , we claim
We conclude the proof by applying Proposition 1. □
Remark 4
If \({M_{h}^{m}}\) is conforming trace space for \({V_{h}^{j}}\) and \({W_{h}^{k}}\), we can approximate terms containing hybrid variables \(\widetilde {u}, \widetilde {v}_{h}\) with the Hσ-norm for interior domain and with the Hϕ-norm for exterior domain. However, it does not improve the convergence rate.
4 Iterative solution
For the solution of the linear system we will iterate on the Schur complement for the trace variable, solving independently in the two sub domains. To justify this split approach we here show that a simple relaxed Jacobi iteration on the two systems will converge. The condition number of the Schur complement can be analysed using the arguments of [7, Section 4].
-
1.
Given \(\widetilde u^{n}\) solve for un+ 1 and λn+ 1 by solving the linear system
$$ A_{h}[(u^{n+1}, \lambda^{n+1}, \widetilde u^{n}), (v,\varphi,0)] = {\int}_{{\Omega}^{-}} f v ~dx. $$ -
2.
Given un+ 1 and λn+ 1, solve for the new trace variable \(\widetilde u^{n+1}\), for σ > 0 and \({ \tau _{h} := \left (\tfrac {\tau _{F}}{h} + \tau _{B} \right )}\),
$$ A_{h}[(u^{n+1}, \lambda^{n+1}, \widetilde u^{n+1}), (0,0,\widetilde v)] + \sigma { \tau_{h}} \left<\widetilde u^{n+1} - \widetilde u^{n},\widetilde v \right>_{\Gamma} = 0. $$
To prove that the iterative algorithm converges we only need to show that if f = 0, un+ 1, \(\widetilde u^{n+1}\) and λn+ 1 all go to zero as \(n \rightarrow \infty \).
We add and subtract \(\widetilde u^{n+1}\) in the first equation and add the second to obtain
Test this equation with \(u^{n+1}, \ \lambda ^{n+1}, \ \widetilde u^{n+1}\) and use coercivity to obtain
Here we used the well-known formula
Considering the terms on the right-hand side and using trace inequality (14) we see that
Using the duality pairing between \(H^{\frac 12}\) and \(H^{-\frac 12}\) followed by the global inverse inequality \(\|\widetilde u^{n+1} - \widetilde u^{n}\|_{H^{\frac 12}({\Gamma })} \leq C_{t} h^{-\frac 12} \|\widetilde u^{n+1} - \widetilde u^{n}\|_{L^{2}({\Gamma })}\) (see [31, Theorem 4.4.3]) and Young’s inequality we have
Finally
Using the once again the telescoping property (30) we see that for
the right-hand sides can all be absorbed in the left-hand side to yield
It follows that as \(N \rightarrow \infty \) un+ 1, \(\widetilde u^{n+1}\) and λn+ 1 all go to zero, since the sum of the left-hand side has to be bounded by the constant of the right-hand side.
5 Numerical experiments
Let \(\boldsymbol {\phi } := [\phi _{1}, \dots , \phi _{j}]^{T}\) be the vector of canonical basis functions of the finite element space \({V_{h}^{j}}\), \(\boldsymbol {\psi } := [\psi _{1}, \dots , \psi _{k}]^{T}\) — the vector of canonical basis functions of \({W_{h}^{k}}\), \(\boldsymbol {\xi } := [\xi _{1}, \dots , \xi _{l}]^{T}\) — the vector of canonical basis functions of \({{\Lambda }_{h}^{l}}\) and \(\boldsymbol {\theta } := [\theta _{1}, \dots , \theta _{m}]^{T}\) be the vector of canonical basis functions of \({M_{h}^{m}}\). We define the following matrices and vector associated with the corresponding linear forms
In the following we present the full definition of these quantities. \(\\ A_{\alpha \beta } = {\int \limits }_{{\Omega }^{-}} \nabla \phi _{\alpha } : \nabla \phi _{\beta } + \varepsilon \phi _{\alpha } \phi _{\beta } dx - \left < \partial _{n} \phi _{\alpha }, \phi _{\beta }\right >_{\Gamma } - \left < \phi _{\alpha } , \ \partial _{n} \phi _{\beta } \right >_{\Gamma } + \tfrac {\tau _{F}}{h} \left < \phi _{\alpha },\ \phi _{\beta } \right >_{\Gamma },\)
Using the above definition the discrete problem (15) can be written in the following matrix form
To solve this system we use the Schur complement and the Steklov-Poincaré operator to eliminate all variables except the hybrid one as follows
In our experiment tests we consider ε = 1 and \(\mathcal {V}_{h}\times {{\Lambda }_{h}^{l}} \times {M_{h}^{m}}\) with j = k = m = 1. The value l varies depending on the geometry of domains considered. We let the trace meshes \(\mathcal {G}_{1}\) and \(\mathcal {G}_{2}\) coincide with the trace mesh of \(\mathcal {T}_{h}\) on Γ. As we know from our experiments as well as the one performed in [2], there is a flexibility with the choice of positive parameter τB, hence for simplicity we use τB = 1.
For our experiments we use two numerical softwares: FEniCS [1] and Bempp [33]. We use the solution of interior and exterior Dirichlet boundary value problems to construct a Schur complement system (31). The solution 𝜃 on Γ of the eliminated system (31) is obtained using the nested conjugate gradient method (CG) [23]. Although one can use direct solvers to solve the interior and exterior Dirichlet boundary value problems, we here used preconditioned iterative solvers suitable for large scale applications. The interior Dirichlet boundary value problem that is a symmetric system associated with bilinear form ah (15) is solved by using FEniCS and CG without and with algebraic multigrid preconditioner. The discrete exterior problem associated with bilinear form bh (15) is not symmetric, however as we shown in Section 3.1, we can apply the Steklov-Poincaré operator to the flux variable and transform the equations into a symmetric system. For clarity of the code, we here simply used the generalised minimal residual method (GMRES) [30] without or with mass matrix preconditioner to solve in Bempp the external Dirichlet boundary value problem. The tolerance of the iterative solvers is chosen to be not greater the 10− 8. A Jupyter notebook demonstrating the functionality used in this paper will be made available at www.bempp.com.
5.1 Choice of parameter τ F
Thanks to Lemma 7 we know that the stabilisation parameter τF in the discrete problem (15) must be large enough to assure coercivity. We start with an experiment showing how the value of the parameter τF influences the convergence and the number of iterations. We consider Ω− as a unit sphere with boundary Γ. We define
It is easy to check that for the unit sphere domain Ω− the above elementary functions are the solution of our problem (1).
Figure 1 shows the error values for different values of τ and j = k = m = l = 1. In this case, Γ is smooth, and so \({W_{h}^{1}} = {{\Lambda }_{h}^{1}}\). In Fig. 1a, we plot in log-log scale the error of the interior solution \(\|u^{-} -u_{h}^{-}\|_{L^{2}({\Omega }^{-})}\) for h ≈ 2− 2 (dashed line with circles), h ≈ 2− 3 (dash-dotted line with diamonds), and h ≈ 2− 4 (solid line with squares).
It can be seen from Fig. 1 that errors stop decreasing when τF is around 10. Furthermore, for τF > 10 the iterations increase with growing τF, hence we fix τF = 10 for the next experiments.
5.2 Spherical subdomain
Let Ω− once again be the unit sphere, Γ its boundary and consider the same exact solution as above.
Figure 2 shows the convergence, CG iteration counts and solving time when τF = 10 and k = l = 1. In Fig. 2a, we plot in log-log scale the error of the interior solution \(\|u^{-} -u_{h}^{-}\|_{L^{2}({\Omega }^{-})}\) (dashed line with circles) and the error of the exterior solutions \(\|u^{+} - u_{h}^{+}\|_{L^{2}({\Gamma })} + \|\lambda ^{+} - \lambda _{h}^{+}\|_{L^{2}({\Gamma })}\) (solid line with squares).
In Fig. 2b, we plot in log-log scale the number of iterations taken by CG to solve the non-preconditioned system associated with exterior problem (dashed line), compared with the preconditioned system (solid line). In addition, Fig. 2c shows the time required by solvers of interior and exterior systems. The interior system is solved by CG with or without algebraic multigrid preconditioner and the exterior system is solved by GMRES with or without mass preconditioner. As we can see in Fig. 2b, the preconditioning reduces both the iteration count and the increase of iterations under refinement. In terms of CPU time needed by the solver, the preconditioned methods reduces the execution time by an order of magnitude compared to the method without preconditioning.
5.3 Cubical subdomain
Let Ω− = (0,1)3 be a cube and we solve the problem (1) with f = 1. We choose τF = 10 and j = k = m = l + 1 = 1, where \({{\Lambda }_{h}^{0}}\) is the space of piecewise constants per element in the trace space. Since the domain has corners we must use an approximation of the flux variable that is discontinuous over corners on the boundary. Therefore we consider the approximation space consisting of functions that are piecewise constant per element, l = 0.
Figure 3a shows the convergence when τF = 10 and j = k = m = l + 1 = 1. In this case, the exact solution is not known; thus, in Fig. 3a, we plot in log-log scale the error between interior solution and exterior \(\frac {\|u_{h}^{-}|_{\Gamma } -u_{h}^{+}\|_{L^{2}({\Gamma })}}{\|u_{h}^{+}\|_{L^{2}({\Gamma })}}\) (solid line).
In Fig. 3, we plot as well in log-log scale the number of iterations and solving time taken by CG to solve the non-preconditioned system (dashed line), compared with the preconditioned system (solid line). Once again preconditioning brings improvement in terms of iteration counts and time taken to solve the problem. The reduction of the iteration count is less significant than in the previous case, however we can see similar improvement in terms of time reduction.
6 Conclusions
We have analysed and demonstrated the effectiveness of Nitsche-type methods for coupling finite element and boundary element formulations. Our approach gives flexibility to choose a continuous or discontinuous finite element space in the FEM solver, hence the interior problem can be solved essentially using any method that allows for the hybridised Nitsche method for interdomain coupling. We are also free to choose the trace variable minimising the coupling degrees of freedom. In this paper, we focus on the technical aspects and analysis to allow for flexibility within our framework. A work demonstrating the applicability for large problems using parallel approach is in preparation.
The method can be extended to other models such as the Helmholtz equations. In this case it is known [36] that the use of impedance interface conditions is advantageous and such an approach can be mimicked in the present framework by letting the stabilisation constant have non-zero imaginary part and depend on the wave number. Formulations of the presented FEM/BEM coupling method to the Helmholtz and Maxwell problems are currently in preparation. For these cases however more effective operator preconditioning techniques for exterior problem are essential, especially for high frequency problems. Despite that, we expect that their implementation will be similar to the presented Laplace case.
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
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All authors were supported by EPSRC grant EP/P01576X/1. Timo Betcke and Erik Burman were also supported by EPSRC grant EP/P012434/1.
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Betcke, T., Bosy, M. & Burman, E. Hybrid coupling of finite element and boundary element methods using Nitsche’s method and the Calderon projection. Numer Algor 91, 997–1019 (2022). https://doi.org/10.1007/s11075-022-01289-9
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DOI: https://doi.org/10.1007/s11075-022-01289-9