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Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations

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Abstract

A conforming discontinuous Galerkin (CDG) finite element method is designed for solving second order non-self adjoint and indefinite elliptic equations. Unlike other discontinuous Galerkin (DG) methods, the numerical trace on the edge/triangle between two elements is not the average of two discontinuous \(P_k\) functions, but a lifted \(P_{k+2}\) function from four (eight in 3D) nearby \(P_k\) functions. While all existing DG methods have the optimal order of convergence, this CDG method has a superconvergence of order two above the optimal order when solving general second order elliptic equations. Due to the superconvergence, a post-process lifts a \(P_k\) CDG solution to a quasi-optimal \(P_{k+2}\) solution on each element. Numerical tests in 2D and 3D are provided confirming the theory.

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Authors and Affiliations

  1. Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    Xiu Ye

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

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  1. Xiu Ye
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  2. Shangyou Zhang
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Correspondence to Shangyou Zhang.

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Communicated by: Jon Wilkening

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Ye, X., Zhang, S. Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations. Adv Comput Math 50, 2 (2024). https://doi.org/10.1007/s10444-023-10100-9

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  • DOI: https://doi.org/10.1007/s10444-023-10100-9

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