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Optimal A Priori Error Estimates for Elliptic Interface Problems: Weak Galerkin Mixed Finite Element Approximations

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Abstract

We consider weak Galerkin mixed finite element approximations of second order elliptic equations on domains that can be described as a union of subdomains or blocks. This paper establishes optimal a priori error estimates for the unknown function u and vector flux \(\varvec{\nabla } \varvec{u}\) in discrete H\(^{\textbf{1}} \) norm with respect to the local regularity of the solution. In addition, high-order convergence rate has been achieved in L\(^{\textbf{2}} \) norm for the unknown function u by using suitable weak Galerkin approximation space of higher degree. We have performed some typical numerical tests to confirm the theoretical findings of the proposed algorithm. Moreover, numerically it is shown that the proposed algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.

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Acknowledgements

Authors are grateful to the anonymous referees for their valuable comments and suggestions.

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  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India

    Raman Kumar & Bhupen Deka

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  1. Raman Kumar
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  2. Bhupen Deka
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Correspondence to Bhupen Deka.

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Kumar, R., Deka, B. Optimal A Priori Error Estimates for Elliptic Interface Problems: Weak Galerkin Mixed Finite Element Approximations. J Sci Comput 97, 27 (2023). https://doi.org/10.1007/s10915-023-02333-z

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  • DOI: https://doi.org/10.1007/s10915-023-02333-z

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