Application of Finite Element Method to Problems with Mixed Boundary Conditions

Abstract

In this study, we explore the effectiveness of the Finite Element Method (FEM) employing linear shape functions to address problems governed by Dirichlet, Neumann, and Robin boundary conditions. We use the derived weak formulation of FEM to solve various types of partial differential equations (PDEs) with mixed boundary conditions. Convergence and stability analyses are carried out to evaluate the performance of this approach, and different types of errors, like absolute error, dissipation, dispersion, and total mean square error, are investigated. This method is applied, and both the exact and approximate solutions are tabulated in three distinct cases: a one-dimensional Burgers-Huxley equation with Dirichlet boundary conditions; a diffusionreaction equation with Neumann boundary conditions; and a uniformly propagating shock problem with Robin boundary conditions. Approximate solutions are compared to exact ones through 2D and 3D graphical representations, and tabular data offers a thorough error analysis. Additionally, error maps provide strong evidence for the accuracy of the suggested approach, demonstrating its capacity to precisely and quickly solve challenging problems with a variety of boundary conditions.

J. Bangladesh Math. Soc. 44.2 (2024) 047–064