An Arbitrary-Order Ultra-Weak Discontinuous Galerkin Method for Two-Dimensional Semilinear Second-Order Elliptic Problems on Cartesian Grids

An Arbitrary-Order Ultra-Weak Discontinuous Galerkin Method for Two-Dimensional Semilinear Second-Order Elliptic Problems on Cartesian Grids

Year:    2022

Author:    Mahboub Baccouch

International Journal of Numerical Analysis and Modeling, Vol. 19 (2022), Iss. 6 : pp. 864–886

Abstract

In this paper, we present and analyze a new ultra-weak discontinuous Galerkin (UWDG) finite element method for two-dimensional semilinear second-order elliptic problems on Cartesian grids. Unlike the traditional local discontinuous Galerkin (LDG) method, the proposed UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a system of equations. The UWDG scheme is presented in details, including the definition of the numerical fluxes, which are necessary to obtain optimal error estimates. The proposed scheme can be made arbitrarily high-order accurate in two-dimensional space. The error estimates of the presented scheme are analyzed. The order of convergence is proved to be $p + 1$ in the $L^2$-norm, when tensor product polynomials of degree at most $p$ and grid size $h$ are used. Several numerical examples are provided to confirm the theoretical results.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/2022-IJNAM-21037

International Journal of Numerical Analysis and Modeling, Vol. 19 (2022), Iss. 6 : pp. 864–886

Published online:    2022-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    23

Keywords:    Ultra-weak discontinuous Galerkin method elliptic problems convergence a priori error estimation.

Author Details

Mahboub Baccouch