Ritz–Galerkin method

In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem (such as a differential equation) to a discrete problem. In principle, it is the equivalent of applying the method of variation of parameters to a function space, by converting the equation to a weak formulation. Typically one then applies some constraints on the function space to characterize the space with a finite set of basis functions. Galerkin's method provides powerful numerical solution to differential equations and modal analysis.

The approach is usually credited to Boris Galerkin but the method was discovered by Walther Ritz, to whom Galerkin refers. Often when referring to a Galerkin method, one also gives the name along with typical approximation methods used, such as Bubnov–Galerkin method (after Ivan Bubnov), Petrov–Galerkin method (after Georgii I. Petrov) or Ritz–Galerkin method (after Walther Ritz).

Examples of Galerkin methods are:

Let us introduce Galerkin's method with an abstract problem posed as a weak formulation on a Hilbert space ${\displaystyle V}$, namely,

Here, ${\displaystyle a(\cdot ,\cdot )}$ is a bilinear form (the exact requirements on ${\displaystyle a(\cdot ,\cdot )}$ will be specified later) and ${\displaystyle f}$ is a bounded linear functional on ${\displaystyle V}$.

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