A $C^0$ finite element method for the biharmonic problem with Navier boundary conditions in a polygonal domain

TL;DR

This paper introduces a novel $C^0$ finite element method for solving the biharmonic equation with Navier boundary conditions in polygonal domains, effectively handling both convex and non-convex cases.

Contribution

It proposes a new decoupling approach involving three Poisson equations and develops an optimal error estimate for the method.

Findings

Method works for convex and non-convex domains.

Achieves optimal error estimates on various meshes.

Numerical tests confirm theoretical results.

Abstract

In this paper, we study the biharmonic equation with the Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the 4th-order problem into a system of Poisson equations. Different from the usual mixed method that leads to two Poisson problems but only applies to convex domains, the proposed decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and non-convex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulted system. In addition, we derive the 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.