PatchMethoden

PatchMethoden, also known as Patch Methods, refers to a category of algorithms used in computational mathematics and computer science for solving systems of equations, particularly differential equations. These methods are characterized by their iterative nature, where a solution is progressively refined by applying corrections or "patches" to an initial guess. The core idea is to start with an approximate solution and then systematically improve it by addressing the error or residual.

A common application of patch methods is in the numerical solution of partial differential equations (PDEs). In this context, a domain is discretized into a grid of points or elements. The PDE is then approximated at these discrete locations. Patch methods work by iteratively updating the solution values at these points. The update process typically involves calculating the residual (the difference between the current approximation and the true solution) and using this residual to compute a correction. This correction is then added to the current approximation, bringing it closer to the true solution.

Various specific algorithms fall under the umbrella of patch methods. These can include iterative solvers like Gauss-Seidel or successive over-relaxation (SOR) for linear systems, which can arise from the discretization of differential equations. More complex methods might involve adaptive mesh refinement, where the grid itself is adjusted dynamically to focus computational effort on regions where the solution is changing rapidly or where the error is high. The effectiveness of a patch method often depends on factors such as the initial guess, the convergence criteria, and the nature of the problem being solved. They are widely used in fields like fluid dynamics, structural analysis, and heat transfer.