A mixed finite element method for a sixth order elliptic problem

TL;DR

This paper introduces a novel mixed finite element method for sixth order elliptic problems, utilizing a saddle point formulation that enables the use of standard $H^1$-conforming elements, with proven error estimates and numerical validation.

Contribution

It presents a new saddle point formulation for sixth order PDEs that allows standard finite element spaces to be used, with proven error bounds and numerical demonstrations.

Findings

Error estimates are established for the proposed method.

Numerical results confirm the theoretical convergence rates.

The approach effectively handles different boundary conditions.

Abstract

We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the $H^1$-conforming Lagrange finite element spaces to approximate the solution. We prove a priori error estimates for our approach. Numerical results are presented for linear and quadratic finite element methods.