A mixed finite element method for a sixth order elliptic problem
This work provides a new numerical approach for solving high-order elliptic PDEs, but it is an incremental extension of existing methods (Ciarlet-Raviart formulation) to a sixth-order problem.
The authors develop a mixed finite element method for a sixth-order elliptic problem using H^1-conforming Lagrange elements, proving a priori error estimates and demonstrating numerical results for linear and quadratic elements.
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.