Subparametric Finite Element method for Some Elliptic Equations

Abstract

A powerful automated high-order unstructured curved mesh generator is proposed in this work with a high-level dynamic programming language Julia. This generator uses higher-order one-sided curved triangular finite elements for the domains having curved borders with parabolic arcs. The proposed approach of the mesh generator can be successfully applied for solving several industrial problems inclusive of photonics with the finite element method. The use of the parabolic arcs method to obtain node relations for the curved geometry enhances the performance of the technique with the subparametric mappings. The mesh generator suggested is based on the prominent Gmsh mesh generator. For all geometry, the presented technique can be implemented. The methodology is applied in this paper to illustrate a few photonic crystal domains. An enriched finite element method is presented to numerically solve the eigenvalue problem on electromagnetic waveguides governed by the Helmholtz equation. In this work, a highly efficient, simple and precise higher order subparametric method was developed using a 2D automated mesh generator performed with JuliaFEM. The transcendence computerized discretization code in Julia is developed for the present work. For curved waveguide structures, meshes with one side and curving higher orders are proposed with triangular elements with parabolic arcs. The technique is shown for distinct waveguide constructions, and the results are compared with the strongest numerical or analytical results available. The results demonstrate that the proposed methodology is effective and accurate for generating finite element simulations for complex structures with black holes and irregular topology due to no curvature loss. This article presents a finding cutoff frequency performed with JuliaFEM an open- source program. Analysis results produced by commercial software are considered for the comparison and show that the calculation results between the two programs do not differ significantly. This procedure can be...