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A numerical method for solving variable coefficient elliptic equation with interfaces. (English) Zbl 1061.65123
Summary: A new second-order accurate numerical method on non-body-fitting grids is proposed for solving the variable coefficient elliptic equation in disjoint subdomains $\Omega^{\pm}$ separated by interfaces $\Gamma$. The variable coefficients, the source term, and hence the solution itself and its derivatives may be discontinuous across the interfaces. Jump conditions in solution and its co-normal derivative at interface are prescribed. Instead of smooth, the interfaces are only required to be Lipschitz continuous as submanifold. A weak formulation is developed, the existence, uniqueness and regularity of the solutions are studied. The numerical method is derived by discretizing the weak formulation. The method is different from traditional finite element methods. Extensive numerical experiments are presented and show that the method is second-order accurate in solution and first-order accurate in its gradient in $L^\infty$ norm if the interface is $C^2$ and solutions are $C^2$ on the closures of the subdomains. The method can handle the problems when the solutions and/or the interfaces are weaker than $C^2$. For example, $u\in H^2(\Omega^{\pm})$, $\Gamma$ is Lipschitz continuous and their singularities coincide, see Example 18 in Section 4. The accuracies of the method under various circumstances are listed in Table 19.

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J25Second order elliptic equations, boundary value problems
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References:
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