Galerkinformuleringen

Galerkinformuleringen, also known as the Galerkin method, is a numerical technique used to solve partial differential equations (PDEs). It was developed by Boris Galerkin in 1915 and is widely used in engineering and applied mathematics. The method is based on the idea of approximating the solution of a PDE by a linear combination of basis functions. These basis functions are chosen such that they satisfy the boundary conditions of the problem.

The Galerkin method involves the following steps:

1. Choose a set of basis functions {φ1, φ2, ..., φn} that satisfy the boundary conditions of the PDE.

2. Approximate the solution u(x) of the PDE as a linear combination of the basis functions: u(x) ≈ ∑ c_i φ_i(x), where c_i are unknown coefficients.

3. Substitute the approximate solution into the PDE and multiply both sides by each basis function φ_i.

4. Integrate both sides of the resulting equations over the domain of the problem.

5. Solve the resulting system of linear equations for the coefficients c_i.

The Galerkin method is particularly useful for solving PDEs that are difficult or impossible to solve analytically. It is also well-suited for problems with complex geometries and boundary conditions. However, the accuracy of the method depends on the choice of basis functions and the number of terms used in the approximation. In practice, the Galerkin method is often combined with other numerical techniques, such as the finite element method, to solve more complex problems.