Numerical Demonstration of Finite Element Convergence for Lagrange Elements in COMSOL Multiphysics

The convergence order of finite elements is related to the polynomial order of the basis functions used on each element, with higher order polynomials yielding better convergence orders. However, two issues can prevent this convergence order from being achieved: the lack of regularity of the PDE solution and the poor approximation of curved boundaries by polygonal meshes. We show studies for Lagrange elements of degrees 1 through 5 applied to the classical test problem of the Poisson equation with Dirichlet boundary condition. We consider this problem in two spatial dimensions with smooth and non-smooth data on domains with polygonal and with curved boundaries. The observed convergence orders in the norm of the error between FEM and PDE solution demonstrate that they are limited by the regularity of the solution and are degraded significantly on domains with non-polygonal boundaries. All numerical tests are carried out with COMSOL Multiphysics.