GalerkinProjektion

GalerkinProjektion is a numerical method used in applied mathematics and engineering to approximate solutions to differential equations. It is a type of variational method, meaning it seeks a solution in a simpler, approximate space that satisfies a weak form of the original equation. The core idea is to project the unknown solution onto a finite-dimensional subspace. This projection is performed by ensuring that the residual, the difference between the approximated solution and the true solution in the differential equation, is orthogonal to the chosen subspace.

The method is named after the Russian mathematician Boris Galerkin. In practice, this orthogonality condition leads to a system of algebraic equations that can be solved computationally. The choice of the finite-dimensional subspace is crucial and often involves using basis functions, such as polynomials or splines. Common examples of Galerkin methods include the Finite Element Method (FEM) and the Discontinuous Galerkin (DG) method, which are widely used in fields like fluid dynamics, solid mechanics, and heat transfer. The accuracy of the approximation generally increases with the dimension of the chosen subspace.