What are the different methods used for solving boundary value problems with finite element method?

In numerical analysis, the finite element method is commonly used to solve boundary value problems. This method involves dividing the problem domain into smaller subdomains called finite elements, and then approximating the solution within each element using a set of basis functions. To solve boundary value problems with the finite element method, several methods can be employed. Here are some of the commonly used methods:

1. Direct Method: This approach involves directly solving the system of equations obtained from the discretization of the problem using finite element basis functions. The system of equations can be solved using techniques such as Gaussian elimination or LU decomposition.

2. Iterative Method: In this method, an initial guess for the solution is made, and then an iterative process is used to refine the solution until convergence is achieved. Examples of iterative methods include the Jacobi method, Gauss-Seidel method, and successive over-relaxation (SOR) method.

3. Penalty Method: The penalty method introduces additional terms in the governing equations to enforce the boundary conditions. These additional terms penalize the violation of the boundary conditions and are gradually increased until the desired accuracy is achieved.

4. Mixed Method: This approach combines the finite element method with other numerical techniques, such as the finite difference method or the finite volume method. It allows for the simultaneous approximation of both the primary variable (e.g., displacement) and its associated flux (e.g., stress or heat flux).

5. Variational Method: The variational method formulates the problem as a minimization of a functional, typically the total potential energy or the action functional. The solution is obtained by minimizing this functional using variational principles, such as the principle of minimum potential energy or the principle of least action.

These are some of the methods commonly used for solving boundary value problems with the finite element method. The choice of method depends on the specific problem and the desired accuracy and efficiency of the solution.