Abstract
In this paper, we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Under assumption that the domain is convex and the mesh is quasi-uniform, a priori error estimate for the error in L2-norm is proved. By duality argument and Oswald interpolation, a posteriori error estimates for the errors in L2-norm and W1,p-seminorm are also obtained. Finally, numerical examples are provided to validate the theoretical analysis.
Highlights
In this article, we consider the following problem−∆u = δx0 in Ω, u = 0 on ∂Ω.(1.1a) (1.1b) where Ω ⊂ Rd (d = 2, 3) is an open, bounded, polygonal or polyhedral domain with Lipschitz boundary ∂Ω, and δx0 is a Dirac measure concentrated at the interior point x0 ∈ Ω
It is worth noting that Oswald interpolation is a very important tool in a posteriori error analysis of HDG methods [4, 5, 14, 17], because it provides a continuous approximation for a discontinuous piecewise polynomial function
We find that the convergence rate O(h) can be achieved
Summary
Hybridizable discontinuous Galerkin method, a priori error estimate, a posteriori error estimate, elliptic equation, Dirac measure. A posteriori error estimator that is efficient and reliable for the error in W 1,p-seminorm is derived in a non-convex Lipschitz polygon, where p ∈ (P Ω, 2) and P Ω > 0 is a real number depending on the largest inner angle of the domain Ω. It is worth noting that Oswald interpolation is a very important tool in a posteriori error analysis of HDG methods [4, 5, 14, 17], because it provides a continuous approximation for a discontinuous piecewise polynomial function.
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