In BEM, the boundary of the domain is discretized into elements, similar to the finite element method (FEM). However, unlike FEM, which requires the discretization of the entire domain, BEM only requires the discretization of the boundary. This reduction in dimensionality can lead to significant computational savings, especially for problems with complex geometries or where the solution is primarily influenced by the boundary conditions.
The BEM involves the formulation of integral equations that relate the boundary values of the solution and its normal derivative. These integral equations are then solved numerically using techniques such as collocation or Galerkin methods. The resulting system of linear equations can be solved using standard numerical methods, providing an approximate solution to the original partial differential equation.
One of the key advantages of BEM is its ability to handle problems with infinite or semi-infinite domains, as well as problems with singularities or discontinuities in the solution. This makes it particularly useful for problems involving cracks, corners, or other geometric features that can be challenging to handle with other numerical methods.
However, BEM also has some limitations. It is generally less efficient for problems with complex internal geometries or where the solution is influenced by the interior of the domain. Additionally, the method can be more sensitive to the choice of discretization and the accuracy of the numerical integration.
In summary, Rajauselementteinä, or boundary element method, is a powerful numerical technique for solving partial differential equations, particularly for problems involving potential fields. Its efficiency and ability to handle complex boundary conditions make it a valuable tool in engineering and applied mathematics.