On the Choice of Robin Parameters for the Optimized Schwarz Method for Domains with Non-Conforming Heterogeneities

Abstract

We consider the solution of $-\nabla \cdot (\nu (x)\nabla u)=0$ by a non-overlapping optimized Schwarz domain decomposition method, where the subdomains do not align with jumps in the coefficient $\nu (x)$. Such a decomposition can be of interest when the jumps are geometrically complex and/or an artifact of the measured data, in which case one would often prefer a simpler decomposition that disregards the location of the discontinuities. For analysis purposes, we focus on a model problem where the diffusivity is piecewise constant, and we analyze the convergence of optimized Schwarz for the two-subdomain case. We consider using either a constant Robin parameter along the whole interface, or a parameter that is scaled proportionally to the local diffusivity. We show that the convergence rate is not robust with respect to the heterogeneity ratio when a constant Robin parameter is used; however, using a scaled Robin parameter restores robustness. We then derive optimal scaling parameter and the corresponding convergence factor. Numerical examples show that this choice also leads to robust convergence behaviour for cases not covered by the analysis.

Appendix: Useful Sobolev-Type Estimates

Lemma 3

([28], Lemma B.5) Let $\phi$ be a basis function associated with a node of an element $K \subset \varOmega \subset \mathbb {R}^n$, Then there exist constants c and C, independent of h, such that

$$\begin{aligned} c h^{n} \le \Vert \phi \Vert _{L^{2}(K)}^{2} \le C h^{n}. \end{aligned}$$

Consequently, for any $u_\varGamma$, the trace of a finite element function on $\varGamma$, there exist constants $c_1$ and $C_1$, independent of h and H, such that

$$\begin{aligned} c_1 h^{n-1} \mathbf {u}_\varGamma ^{T} \mathbf {u}_\varGamma \le \Vert u_\varGamma \Vert _{L^2(\varGamma )}^{2} \le C_1 h^{n-1} \mathbf {u}_\varGamma ^{T} \mathbf {u}_\varGamma . \end{aligned}$$

Lemma 4

([28], Lemma A.14 and Corollary A.15) Let $\varOmega$ be a Lipschitz continuous domain with diameter H. Let $\varGamma \in \partial \varOmega$ have nonvanishing measure. If $u\in H^1(\varOmega )$ vanishes on $\varGamma$, then there exists a constant C, independent of H, such that

$$\begin{aligned} \Vert {u} \Vert _{L^2(\varOmega )}^2 \le C^2 H^2 \Big \vert {u}\Big \vert _{H^1(\varOmega )}^2. \end{aligned}$$

Moreover, if we define

$$\begin{aligned} \Vert {u} \Vert _{H_s^{1}(\varOmega )}^{2}:=\Big \vert {u}\Big \vert _{H^{1}(\varOmega )}^{2}+\frac{1}{H^{2}}\Vert {u} \Vert _{L^{2}(\varOmega )}^{2} \end{aligned}$$

as the scaled $H^1$-norm, then we obtain the following estimate:

$$\begin{aligned} \Vert {u} \Vert _{H_s^{1}(\varOmega )}^{2}\le (1+C^2)\Big \vert {u}\Big \vert _{H^{1}(\varOmega )}^{2}. \end{aligned}$$

Lemma 5

Let $u_\varGamma$ be the trace of a finite element function on $\varGamma$, and let $\mathbf {u}_\varGamma$ be the vector corresponding to $u_\varGamma$. Let $A_1$ be the stiffness matrix associated with the standard Laplacian bilinear form, and $S_1$ be its Schur complement after the interior nodes have been eliminated. Then there exist constants c and C, independent of h and H, such that the following sharp estimate holds:

$$\begin{aligned} cH^{-1}\Vert {u_\varGamma } \Vert _{L^2(\varGamma )}^2 \le \mathbf {u}_\varGamma ^TS_1\mathbf {u}_\varGamma \le C h^{-1}\Vert {u_\varGamma } \Vert _{L^2(\varGamma )}^2, \end{aligned}$$

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where h is the diameter of the finite element, and H is the diameter of the subdomain. Consequently, there exists a constant $C'$, independent of h and H, such that

$$\begin{aligned} \kappa (S_1)<C'\frac{H}{h}. \end{aligned}$$

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Proof

The proof is based on Lemma 4.11 in [28], where the authors have derived the upper bound. For the lower bound, they have also proved that

$$\begin{aligned} H\Vert {u} \Vert _{L^{2}(\varGamma )}^{2} \le H^{2}\Vert u\Vert _{H^{1 / 2}(\varGamma )}^{2} \le C_{t}^{2}\left( H^{2}\Big \vert {u}\Big \vert _{H^{1}(\varOmega _{i})}^{2}+\Vert {u} \Vert _{L^{2}(\varOmega _{i})}^{2}\right) , \end{aligned}$$

where $C_t$ is a constant independent of h and H. Simplification of the above inequality gives

$$\begin{aligned} H^{-1}\Vert {u} \Vert _{L^{2}(\varGamma )}^{2} \le C_{t}^{2}\left( \;\Big \vert {u}\Big \vert _{H^{1}(\varOmega _{i})}^{2}+\frac{1}{H^2}\Vert {u} \Vert _{L^{2}(\varOmega _{i})}^{2}\right) =C_{t}^{2}\Vert {u} \Vert _{H_s^1(\varOmega _i)}^2, \end{aligned}$$

where we have used the definition of the scaled $H^1$-norm (see also (4.4) in [28]). By Lemma 4, we obtain

$$\begin{aligned} H^{-1}\Vert {u} \Vert _{L^{2}(\varGamma )}^{2} \le C \Big \vert {u}\Big \vert _{H^1(\varOmega _i)}^2. \end{aligned}$$

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Since $\mathbf {u}_\varGamma ^TS_1\mathbf {u}_\varGamma = \Big \vert {u}\Big \vert _{H^1(\varOmega _i)}^2$, actually we have finished the proof for the lower bound.

The condition number estimate is then a direct result from the spectral estimates. Note that $\kappa (S_1)$ is none other than the condition number of the Schur complement matrix for the Laplace operator. $\square$

Lemma 6

([28], Lemma 4.10) Let $\varOmega$ be a domain, and let a finite element trace function $u_\varGamma$ be given on the boundary $\varGamma =\partial \varOmega$. Let $\mathbf {u}=\mathscr {H}_1(\mathbf {u}_\varGamma )$ be the discrete 1-harmonic extension of $u_\varGamma$ into $\varOmega$. Then there exist constants c and C, independent of h and H, such that

$$\begin{aligned} c\left| u_{\varGamma }\right| _{H^{1 / 2}(\varGamma )}^{2} \le |u|_{H^{1}\left( \varOmega _{1}\right) }^{2} \le C\left| u_{\varGamma }\right| _{H^{1 / 2}(\varGamma )}^{2}. \end{aligned}$$

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Lemma 7

([11], Lemma 4.5) Let $\varOmega$ be a domain, and let a finite element trace function $u_\varGamma$ be given on the boundary $\varGamma =\partial \varOmega$. Then there exist a constant C, independent of h and H, such that

$$\begin{aligned} \Big \vert {u_{\varGamma }}\Big \vert _{H^{1/2}(\varGamma )}^{2} \le \frac{C}{h}\Vert {u_{\varGamma }} \Vert _{L^{2}(\varGamma )}^{2}. \end{aligned}$$

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Lemma 8

([28], Lemma A.17) Let $\varOmega$ be a bounded Lipschitz continuous polygon or polyhedron with diameter H. Then for any $u\in H^{1/2}(\partial \varOmega )$ such that the measure of the set $\{x\in \partial \varOmega ; u(x)=0.\}$ is non-zero, there exists a constant C, independent of h, such that

$$\begin{aligned} \Vert {u} \Vert _{L^2(\partial \varOmega )}^2\le CH\Big \vert {u}\Big \vert _{H^{1/2}(\partial \varOmega )}^2. \end{aligned}$$

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