A <i>C</i>0 finite element method for the biharmonic problem with Navier boundary conditions in a polygonal domain

Hengguang Li, Peimeng Yin, Zhimin Zhang

2022IMA Journal of Numerical Analysis10 citationsDOI

Abstract

Abstract In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.

Topics & Concepts

Biharmonic equationMathematicsPolygon meshPoisson's equationFinite element methodMathematical analysisDomain decomposition methodsBoundary value problemRegular polygonBoundary (topology)Domain (mathematical analysis)Poisson distributionApplied mathematicsGeometryStatisticsThermodynamicsPhysicsAdvanced Numerical Methods in Computational MathematicsAdvanced Mathematical Modeling in EngineeringNumerical methods in engineering