Solution of the Elliptic Interface Problem by a Hybrid Mixed Finite Element Method

Yuhan Wang, Peiyao Wang, Rongpei Zhang () and Jia Liu
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Yuhan Wang: School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China
Peiyao Wang: School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China
Rongpei Zhang: School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China
Jia Liu: School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 522000, China

Mathematics, 2024, vol. 12, issue 12, 1-11

Abstract: This paper addresses the elliptic interface problem involving jump conditions across the interface. We propose a hybrid mixed finite element method on the triangulation where the interfaces are aligned with the mesh. The second-order elliptic equation is initially decomposed into two equations by introducing a gradient term. Subsequently, weak formulations are applied to these equations. Scheme continuity is enforced using the Lagrange multiplier technique. Finally, we derive an explicit formula for the entries of the matrix equation representing Lagrange multiplier unknowns resulting from hybridization. The method yields approximations of all variables, including the solution and gradient, with optimal order. Furthermore, the matrix representing the final linear algebra systems is not only symmetric but also positive definite. Numerical examples convincingly demonstrate the effectiveness of the hybrid mixed finite element method in addressing elliptic interface problems.

Keywords: hybrid mixed finite element method; discontinuous coefficient; elliptic interface problem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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