What are the different methods used for solving integral equations with finite element method?

There are several methods used for solving integral equations with the finite element method. Some of the commonly used methods include:

1. Galerkin method: This is the most widely used method for solving integral equations with the finite element method. In this method, the integral equation is approximated by a system of algebraic equations using a set of basis functions. The unknown coefficients of the basis functions are then determined by minimizing the residual error.

2. Collocation method: In this method, the integral equation is approximated by a set of discrete equations at specific points in the domain. These discrete equations are obtained by evaluating the integral equation at the collocation points. The unknown coefficients are then determined by solving the resulting system of algebraic equations.

3. Boundary element method: This method is particularly useful for solving integral equations defined on the boundary of a domain. In this method, the integral equation is transformed into a boundary integral equation, where the unknowns are defined only on the boundary. The boundary integral equation is then discretized using the finite element method, and the unknown coefficients are determined by solving the resulting system of algebraic equations.

4. Dual reciprocity method: This method is based on the idea of representing the solution of the integral equation as a linear combination of known functions, called the dual basis functions. The unknown coefficients of the dual basis functions are determined by solving a system of algebraic equations obtained by applying the integral equation to the dual basis functions.

5. Trefftz method: This method is based on the idea of representing the solution of the integral equation as a linear combination of known functions, called the Trefftz functions. The unknown coefficients of the Trefftz functions are determined by solving a system of algebraic equations obtained by applying the integral equation to the Trefftz functions.

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