Computational experiments using nonconforming finite elements

Abstract

Initially, the conforming finite element method is explained. The steps of the finite element method (begin with a second-order differential equation, multiply by a test function, put in integral form, produce a weak formulation by integrating by parts, reach a system of equations, and solve for unknowns) are discussed for a simple univariate case, general univariate case, and Poisson’s equation^1.

A nonconforming finite element method is then used to solve the two-dimensional Poisson's equation. A matlab code, which may be found in the Appendix section A3 , solves the problem on Ù = [—1,1] x [0,1], where Ù1 = [—1.0] x [0,1] and Ù2 = [0,1] X [0,1]. The approximation on the boundary Ã12 =Ù1Ù2 is not truly continuous, but is “weakly" continuous. The result is a discontinuous approximation to a smooth function that demonstrates the feasibility of solving Poisson's equation by combining two separately meshed regions and enforcing "weak" continuity across the boundary.