Assessment of Multigrid Schemes with Fixed Patterns for a Two-Dimensional Heat Transfer Problem

Heat transfer problems and their solutions are of critical importance in almost all areas of engineering and technology, while many real-world problems are inherently three-dimensional. Simplifying them to 2D models offers practical advantages with reasonable models. With being the fundamental class of problem of heat transfer, the 2D thermal diffusion problem was selected for the study. Multigrid methods were referred to as standing out in terms of cost reduction while keeping solution accuracy. The effectiveness of multigrid methods employing fixed pattern schemes was subject to investigation. To set up numerical experimentation, an authentic code generation effort was given that implements a basic finite volume method, intergrid operations and iterative solvers. A reference case with an analytical Laplace solution was selected and properly validated by the results. A variety of multigrid schemes with fixed patterns were explored around parameters, iterations per sweep and maximum coarsening level. Results were compiled on the focal points of cost and performance. A comparison of the direct iterative methods with multigrid schemes proved the effectiveness of multigrid schemes. With the assessment of the cost and performance outcomes, it is concluded that any multigrid scheme should visit the maximum coarsest level possible while keeping a minimum number of iterations on each grid resolution.