An interactive code for automatic (h- and p-versions) mesh generation and the solution of one-dimensional Bio-heat Equation

Abstract

In this study, first, we develop an algorithm for the automatic mesh generation employing the nth order one-dimensional finite elements, preparing the elements data, and forming the element connectivity. Secondly, we modify the techniques for evaluating the integrals needed to compute the components of the element matrices and improve the assembly process to form the global matrices. Then, we develop an interactive computer code incorporating all the algorithms and techniques for obtaining the solution of the one-dimensional Bio-heat equation. The code requires only two basic inputs, the order and the total number of the elements in addition to the endpoints of the domain [a, b] for generating the (h- and p-version) meshes and accomplishing the other necessary computations. Since the Bio-heat equation is in the form of a general second-order ordinary BVP, one can use the code conveniently to compute the solutions of the other boundary value problems by providing the actual coefficients of the differential equations. We extensively investigated the performance of the h- and p-methods by calculating the root-mean squares of the errors of the solutions. Finally, for the clarity and reference, we present the solutions of the Bio-heat equation for the four (h- and p- versions) meshes with the corresponding errors.