Approximation of the Boundary in the Finite Element Solution of Fourth Order Problems

Abstract

A procedure is presented to handle curved boundaries in the finite element solution of fourth order elliptic boundary value problems over bounded regions $\Omega $ of the plane. The region $\Omega $ is triangulated with triangles with at most one curved side along the boundary. Finite elements are constructed on the curved boundary triangles which can be pieced together with conforming finite elements on interior triangles to obtain trial functions which are globally in $C^1$. Error estimates are derived in the energy norm and the effect of numerical integration is analyzed.