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A new multiscale finite element method for high-contrast elliptic interface problems. (English) Zbl 1202.65154
The paper presents a multiscale finite element method (FEM) able to capture more accurately solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces. The following classical problem is treated with this method: \[ \int_{\Omega}\mathcal{A}(x)\nabla u(x)\cdot\nabla v(x)dx=\int_{\Omega}F(x)v(x)dx,\qquad v\in H_{0}^{1}(\Omega), \] where the solution \(u\in H^{1}(\Omega)\) satisfies a Dirichlet condition on \(\partial\Omega \) and \(F\) is given on the bounded domain \(\Omega\subset\mathbb{R}^{2}\). The coefficient \(\mathcal{A}\) will be allowed to jump across a finite number of smooth interior interfaces, the boundaries of \(\Omega_{1},\dots,\Omega_{m};\Omega_{0} = \Omega\setminus\bigcup_{i=1}^{m}\Omega_{i}.\) \(\mathcal{A}\) is piecewise constant and \(\mathcal{A}_{min}=\min \mathcal{A}|_{\Omega_{i}}\). Let
\[ \alpha(x)=\frac{1}{\mathcal{A}_{min}}\mathcal{A}(x),\quad f(x)=\frac{1}{\mathcal{A}_{min}}F(x). \] Then \(\alpha\) is piecewise constant and two “high contrast” cases involving special multiscale nodal bases on a coarse quasiuniform triangular mesh \(\tau_{h}\) are considered:
\[ \begin{aligned}&\text{Case I:}\quad \min_{i=1,\dots,m}\alpha_{i}\rightarrow\infty,\quad \alpha_{0}=1, \\ &\text{Case II:}\quad\alpha_{i}\rightarrow\infty,\quad \max_{i=1,\dots,m}\alpha_{i}\leq K. \end{aligned} \]
\(\alpha_{i}\) are the restrictions to \(\Omega_{i}\) of \(\alpha\). For the resulting FEM multiscale solution \(u_{h}^{MS}\) the authors obtain the following estimates:
\[ |u-u_{h}^{MS}|_{H^{1}(\Omega)}\leq Ch[h|f|^{2}_{H^{\frac{1}{2}}(\Omega)}+\|f\|^{2}_{L_{2}(\Omega)}]^{\frac{1}{2}} \] and
\[ |u-u_{h}^{MS}|_{L_{2}(\Omega)}\leq Ch^{2}[h|f|^{2}_{H^{\frac{1}{2}}(\Omega)}+\|f\|^{2}_{L_{2}(\Omega)}]^{\frac{1}{2}}. \]

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
65N15 Error bounds for boundary value problems involving PDEs
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