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On the Uniform Convergence of the Weak Galerkin Finite Element Method for a Singularly-Perturbed Biharmonic Equation

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Abstract

For the biharmonic equation or this singularly-perturbed biharmonic equation, lower order nonconforming finite elements are usually used. It is difficult to construct high order \(C^1\) conforming, or nonconforming elements, especially in 3D. A family of any quadratic or higher order weak Galerkin finite elements is constructed on 2D polygonal grids and 3D polyhedral grids for solving the singularly-perturbed biharmonic equation. The optimal order of convergence, up to any order the smooth solution can have, is proved for this method, in a discrete \(H^2\) norm. Under a full elliptic regularity \(H^4\) assumption, the \(L^2\) convergence achieves the optimal order as well, in 2D and 3D. Numerical tests are presented verifying the theory.

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Correspondence to Shangyou Zhang.

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The first author was supported in part by National Natural Science Foundation of China (Grant No. 11571026) and also supported by Beijing Natural Science Foundation (Grant No. 1192003).

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Cui, M., Zhang, S. On the Uniform Convergence of the Weak Galerkin Finite Element Method for a Singularly-Perturbed Biharmonic Equation. J Sci Comput 82, 5 (2020). https://doi.org/10.1007/s10915-019-01120-z

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  • DOI: https://doi.org/10.1007/s10915-019-01120-z

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