Numerical Solution of Parabolic Partial Differential Equation by Using Finite Element Method

Abstract

Partial differential equations (PDEs) are used in the real world to model physical phe- nomena such as heat, wave, Laplace, and Poisson equations. For regular shape domains, the heat equation can be solved analytically; however, for irregular domains, the computation of the solu- tion is difficult and numerical methods like Finite Difference Method (FDM) and Finite Element Method (FEM) can be used. FEM provides approximate values at discrete points in the domain. It breaks down a large problem into smaller finite elements. These element’s equations are combined into a system representing the whole problem. We show the comparison between analytic solution, solutions by FDM and FEM. The impact of heat on the material is examined at various positions and multiple positions. We compare the analytical and numerical (by FEM) solution considering several homogeneous materials with various diffusivity values (α). Finally, the simulation results of different non-homogeneous materials were compared. Science and engineering fields that use heat equations can be evaluated using the numerical method applied here.