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Error analysis of a weak Galerkin finite element method for two-parameter singularly perturbed differential equations in the energy and balanced norms

Published: 15 March 2023 Publication History

Highlights

In this paper, the weak Galerkin finite element method is studied for a singularly perturbed problem with two parameters. On a piecewise uniform Shishkin mesh, we prove uniform convergence of the weak Galerkin finite element method with piecewise higher order discontinuous polynomials.
One important feature of the WG-FEM is that it is possible to eliminate the element basis functions on each local interval by static condensation with the help of the weak derivatives and using the basis functions in the element and element boundary separately. Then, the resulting global coupled linear system will only involve the degrees of freedom on the skeleton of the mesh. Therefore, the degrees of freedom in the WG-FEM is comparable to the standard finite element method and the WG-FEM has much less unknown than the discontinuous Galerkin methods.
Using a special interpolation operator and its good features on anisotropic mesh, we established a robust uniform optimal convergence in the corresponding energy norm. Moreover, we presented error analysis in a stronger balanced norm with a locally defined L2-projection for the first time.
Extensive numerical experiments were carried out to support the theoretical findings in the paper.

Abstract

A weak Galerkin finite element method is proposed for solving singularly perturbed problems with two parameters. A robust uniform optimal convergence has been proved in the corresponding energy and a stronger balanced norms using piecewise higher order discontinuous polynomials on a piecewise uniform Shishkin mesh. Finally, we give some numerical experiments to support theoretical results.

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  1. Error analysis of a weak Galerkin finite element method for two-parameter singularly perturbed differential equations in the energy and balanced norms
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              Published In

              cover image Applied Mathematics and Computation
              Applied Mathematics and Computation  Volume 441, Issue C
              Mar 2023
              498 pages

              Publisher

              Elsevier Science Inc.

              United States

              Publication History

              Published: 15 March 2023

              Author Tags

              1. Singularly perturbed boundary value problems
              2. Two-parameter differential equations
              3. Weak galerkin finite element method
              4. Shishkin mesh
              5. Uniform convergence
              6. Balanced norm

              Author Tags

              1. 65L10
              2. 65L11
              3. 65L60

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