Numerical Analysis Questions Medium
There are several methods used for solving partial differential equations (PDEs) with the finite element method (FEM). Some of the commonly used methods are:
1. Galerkin method: This is the most widely used method in FEM. It involves multiplying the PDE by a weight function and integrating over the domain. The weight function is chosen to satisfy certain properties, such as being continuous and having compact support. The resulting equation is then discretized using a finite element basis, and the unknowns are solved for using linear algebra techniques.
2. Petrov-Galerkin method: This method is similar to the Galerkin method, but it uses a different weight function for the test functions than for the trial functions. This can lead to improved stability and accuracy for certain types of PDEs.
3. Least squares method: In this method, the PDE is transformed into a system of equations by minimizing the residual of the PDE in a least squares sense. This can lead to improved accuracy and stability, especially for PDEs with strong boundary conditions.
4. Mixed methods: These methods involve introducing additional unknowns, such as the flux or the gradient of the solution, to the problem. This can lead to improved accuracy and stability, especially for PDEs with mixed boundary conditions or PDEs that have a natural interpretation in terms of fluxes or gradients.
5. Discontinuous Galerkin method: This method allows for discontinuities in the solution across element boundaries. It uses different basis functions on each element and introduces numerical fluxes to enforce continuity and conservation across element boundaries. This method is particularly useful for problems with shocks or other types of discontinuities.
These are just a few of the methods used for solving PDEs with FEM. The choice of method depends on the specific problem being solved and the desired properties of the solution, such as accuracy, stability, and computational efficiency.