Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces

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Abstract
This paper considers the finite-element approximation of the elliptic interface problem: -▽·(σ▽u) + cu = f in Ω ⊂ Rn (n = 2 or 3), with u = 0 on ∂Ω, where σ is discontinuous across a smooth surface Γ in the interior of Ω. First we show that, if the mesh is isoparametrically fitted to Γ using simplicial elements of degree k - 1, with k ≥ 2, then the standard Galerkin method achieves the optimal rate of convergence in the H1 and L2 norms over the approximations Ωl4 of Ωl where Ω ≡ Ωl ∪ Γ ∪ Ω2. Second, since it may be computationally inconvenient to fit the mesh to Γ, we analyse a fully practical piecewise linear approximation of a related penalized problem, as introduced by Babuska (1970), based on a mesh that is independent of Γ. We show that, by choosing the penalty parameter appropriately, this approximation converges to u at the optimal rate in the H1 norm over Ωl4 and in the L2 norm over any interior domain Ωl* satisfying Ωl* Ωl** Ωl4 for some domain Ωl**.

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