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Abstract An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this article, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual‐baseda posteriorierror estimator, an adaptive algorithm is proposed together with its convergence and quasi‐optimality proved for the lowest order case. The primary tool is to bridge the connection between the modified weak Galerkin method and the Crouzeix–Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free. Numerical results are presented to support the theoretical results.
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This content will become publicly available on December 15, 2026
A coupled HDG/DG method for porous media with conducting/sealing faults
We introduce and analyze a coupled hybridizable discontinuous Galerkin/discontinuous Galerkin (HDG/DG) method for porous media in which we allow fully and partly immersed faults, and faults that separate the domain into two disjoint subdomains. We prove well-posedness and present an a priori error analysis of the discretization. Numerical examples verify our analysis.
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- Award ID(s):
- 2110781
- PAR ID:
- 10645441
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Computers mathematics with applications
- ISSN:
- 0898-1221
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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