## Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces.(English)Zbl 0629.65118

The paper considers the finite element approximation of the elliptic interface problem: $$-\nabla (\sigma \nabla u)+cu=f$$ in $$\Omega \subset {\mathbb{R}}^ n$$, $$n=2$$ or 3, with $$u=0$$ on $$\partial \Omega$$, where $$\sigma$$ is discontinuous across a smooth interface $$\Gamma\subset \Omega$$. First it is shown that, if the mesh is isoparametrically fitted to $$\Gamma$$ using simplicial elements of degree k-1, $$k\geq 2$$, then the standard Galerkin method achieves the optimal rate of convergence. Second, since it may be computationally inconvenient to fit the mesh to $$\Gamma$$ the authors analyze a piecewise linear approximation of a related penalized problem, as introduced by I. Babuška [Computing 5, 207-213 (1970; Zbl 0199.506)], based on a mesh that is independent of $$\Gamma$$. The optimal rate of convergence is shown here, as well.
Reviewer: P.Burda

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 35R05 PDEs with low regular coefficients and/or low regular data

Zbl 0199.506
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