Automating the Higher Order Finite Element Method using the Subparametric Transformations for Elliptic Partial Differential Equations

Abstract

A novel, simple, efficient and accurate automated subparametric higher-order finite element method for the computation of solutions for elliptic PDEs over 2D and 3D domains is offered in this thesis. This computational technique is accurate owing to the fact that higher-order finite elements are employed. Its efficiency can be witnessed in the drastic reduction of the computational time which has been achieved by the use of the subparametric transformations with parabolic arcs as the degree of the Jacobian is less for each higher-order element compared to the conventional higher-order finite element method. This work thus presents a simple and systematic process to automate the higher-order finite newlineelement method for obtaining efficient and accurate solutions of some elliptic PDEs newlineby utilizing the best discretization procedure, the finest quadrature rule and an excellent subparametric finite element algorithm.Mesh generation is an important prerequisite for finite element method and hence, prior to automating this technique, high-quality higher-order automated mesh generator MATLAB codes are developed. These mesh generators are based on a simple and most popular linear mesh generator distmesh in MATLAB. The nodal relations for the finite elements are derived using the powerful subparametric point transformations to improve the efficiency of the technique. This approach utilizes up to sextic-order triangular elements for 2D domains and up to quartic-order tetrahedral elements for 3D domains. The meshes with one-sided curved higher-order triangular elements are presented with parabolic arcs for curved geometries in 2D. The meshes with one face curved higher-order tetrahedral elements are presented with parabolic arcs for curved geometries in 3D. For regular geometries newlinewith sharp edges having singularities, the utilization of unstructured higher-order newlinefinite element meshes with refinement for the technique is proposed..