Galerkin method

The Galerkin method (also known Galerkin approach, according to Boris Galerkin ) is a numerical method for the approximate solution of operator equations, such as partial differential equations. It represents the most common form, the "method of weighted residuals ", wherein the resulting residue of an approximate solution is minimized.

Abridged version

Rayleigh and Walther Ritz variational problems in the desired function as a linear combination of basis functions have recognized and thus returned to the variational problem on a common problem of optimizing a function of parameters.

For the solution of an operator equation

The unknown function can also be added, for instance as

Which substituted into the operator equation on the left side of the equal sign is a dependent of the coefficient function is obtained. According to the weighted residual method is chosen, the free coefficients so that they function in the space spanned by certain basic functions, disappears, that is orthogonal to these basic functions. Thus we obtain the following equations to determine the

Which, if the operator is linear, may be a linear system of equations. For obtaining a point - collocation method for the Galerkin method, which is attributed in Russian books also Ivan Grigoryevich Bubnov, so there is, Bubnov - Galerkin method.

More detailed representation

Method

The residual is distributed in the considered area. It is weighted with appropriate weighting functions, hence the term " weighted residuals ". The integral of the weighted residual over the area should be as small as possible or better yet disappear completely. The weighting functions have parameters whose number corresponds to the number of degrees of freedom of the system. They lead to the same number of equations and thus to the same large system of equations which is known from the finite element method. In the Galerkin method, the weighting functions are identical to the basis functions in the elements.

Example

Be a differential operator. Wanted is the solution of the differential equation

With a predetermined function and additional constraints for. For this purpose, an approximate solution for recognized as a linear combination of basis functions from a function space:

With even coefficients to be determined. The function is not yet fulfilled in general, the differential equation (1), there remains a residual

In the space an inner product is defined, for this is true, that is, if for all functions. The inner product is often defined as

Often you can not determine the exact solution for the disappearing for each test function (and thus the residual as well), but only an approximate solution for the inner product of the residual with a set of selected linearly independent " weight functions " disappears:

In the Galerkin method are used as weighting functions just the basic functions of selected, so that there is a system of equations for the coefficients.

Field of application

The Galerkin method is applicable when no natural extremal principle for the solution of the differential equation exists. It is thus a basis of the finite element method and extending their applicability to include other physical problems ( continuum problems ) that do not possess such a natural extremal principle. Examples include stationary or unsteady flows. A natural extremal (natural variational principle ), however, there is for example the case of mechanical problems of solid mechanics, where the energy content must have a minimum.

After Zienkiewicz the Galerkin solution is identical to a natural variation in solution or, at least as interpreted. The finite element method (FEM ) is a special Ritz- Galerkin method.