A TWELVE NODED FINITE ELEMENT APPROXIMATION TO 2D-POISSON EQUATIONS WITH A DIRAC LINE SOURCE

This paper presents the finite element approach to solving the Poisson equation. Using Dirichlet boundary conditions in a two-dimensional polygonal region, the polygon to be discretized is made up of twelve-noded quadrilateral structured meshes. To arrive at a numerical solution, the smaller components must first be solved, and the partial answers must then be combined to provide a solution for the complete mesh. The problem finds applications in various physical domains, such as fluid dynamics, heat conduction, electrostatics, and gravitational potential. However, due to the intricate nature of the domains, which include reentrant corners, fractures, and discontinuities in the solution along the borders, it can be challenging to find exact solutions to these problems. As a result, we propose using the MAPLE-18 program to provide numerical results that corroborate our theoretical conclusions and to suggest a twelve-noded quadrilateral mesh approach that facilitates the solution of the problem, the performance of the Galerkin weighted finite element technique on the generic polygonal domain is demonstrated numerically by use of twelve noded quadrilateral mesh.