Numerical Demonstration of Finite Element Convergence for Lagrange Elements in COMSOL Multiphysics
Permanent Link
Author/Creator
Author/Creator ORCID
Date
Department
Program
Citation of Original Publication
Gobbert, M., and S. Yang. "Numerical Demonstration of Finite Element Convergence for Lagrange Elements in COMSOL Multiphysics." in Proceedings of the COMSOL Conference 2008 Boston. https://www.comsol.com/paper/numerical-demonstration-of-finite-element-convergence-for-lagrange-elements-in-comsol-multiphysics-4965.
Rights
This item is likely protected under Title 17 of the U.S. Copyright Law. Unless on a Creative Commons license, for uses protected by Copyright Law, contact the copyright holder or the author.
Subjects
Abstract
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.