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.
Contents
- A problem in weak formulation
- Galerkin Dimension Reduction
- Galerkin orthogonality
- Matrix form
- Symmetry of the matrix
- Analysis of Galerkin methods
- Well posedness of the Galerkin equation
- Quasi best approximation Cas lemma
- Proof
- References
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:
A problem in weak formulation
Let us introduce Galerkin's method with an abstract problem posed as a weak formulation on a Hilbert space
Here,
Galerkin Dimension Reduction
Choose a subspace
We call this the Galerkin equation. Notice that the equation has remained unchanged and only the spaces have changed. Reducing the problem to a finite-dimensional vector subspace allows us to numerically compute
Galerkin orthogonality
The key property of the Galerkin approach is that the error is orthogonal to the chosen subspaces. Since
Matrix form
Since the aim of Galerkin's method is the production of a linear system of equations, we build its matrix form, which can be used to compute the solution by a computer program.
Let
We expand
This previous equation is actually a linear system of equations
Symmetry of the matrix
Due to the definition of the matrix entries, the matrix of the Galerkin equation is symmetric if and only if the bilinear form
Analysis of Galerkin methods
Here, we will restrict ourselves to symmetric bilinear forms, that is
While this is not really a restriction of Galerkin methods, the application of the standard theory becomes much simpler. Furthermore, a Petrov–Galerkin method may be required in the nonsymmetric case.
The analysis of these methods proceeds in two steps. First, we will show that the Galerkin equation is a well-posed problem in the sense of Hadamard and therefore admits a unique solution. In the second step, we study the quality of approximation of the Galerkin solution
The analysis will mostly rest on two properties of the bilinear form, namely
By the Lax-Milgram theorem (see weak formulation), these two conditions imply well-posedness of the original problem in weak formulation. All norms in the following sections will be norms for which the above inequalities hold (these norms are often called an energy norm).
Well-posedness of the Galerkin equation
Since
Quasi-best approximation (Céa's lemma)
The error
This means, that up to the constant
Proof
Since the proof is very simple and the basic principle behind all Galerkin methods, we include it here: by ellipticity and boundedness of the bilinear form (inequalities) and Galerkin orthogonality (equals sign in the middle), we have for arbitrary
Dividing by