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High order discontinuous Galerkin methods for elliptic problems on surfaces. (English) Zbl 1314.65146

Summary: We derive and analyze high order discontinuous Galerkin methods for second order elliptic problems on implicitly defined surfaces in \(\mathbb R^3\). This is done by carefully adapting the unified discontinuous Galerkin framework of D. N. Arnold et al. [ibid. 39, No. 5, 1749–1779 (2002; Zbl 1008.65080)] on a triangulated surface approximating the smooth surface. We prove optimal error estimates in both a (mesh dependent) energy and \(L^2\) norms. Numerical results validating our theoretical estimates are also presented.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
58J05 Elliptic equations on manifolds, general theory
65N15 Error bounds for boundary value problems involving PDEs
35J99 Elliptic equations and elliptic systems

Citations:

Zbl 1008.65080
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References:

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