Numerical Analysis Questions Medium
The concept of numerical solutions of integral equations with the finite element method involves approximating the solution of an integral equation by discretizing the domain into smaller subdomains or elements.
In the finite element method, the integral equation is transformed into a system of algebraic equations by using a set of basis functions defined over each element. These basis functions are typically piecewise continuous functions that are defined over each element and satisfy certain properties, such as being continuous and differentiable.
The integral equation is then approximated by a linear combination of these basis functions, where the coefficients of the linear combination are the unknowns to be determined. This approximation is valid within each element and is referred to as the local approximation.
By applying the Galerkin method, which involves multiplying the integral equation by a test function and integrating over each element, a system of algebraic equations is obtained. This system of equations relates the unknown coefficients to the values of the test functions and the known data of the integral equation.
The resulting system of equations is then solved numerically using various techniques, such as Gaussian elimination or iterative methods, to obtain the values of the unknown coefficients. Once the coefficients are determined, the approximate solution of the integral equation can be reconstructed by combining the local approximations over each element.
The finite element method allows for the numerical solution of integral equations by providing a flexible and efficient approach to handle complex geometries and boundary conditions. It also allows for the incorporation of additional constraints and conditions, such as boundary conditions or material properties, into the numerical solution.
Overall, the concept of numerical solutions of integral equations with the finite element method involves discretizing the domain, approximating the solution using basis functions, and solving the resulting system of equations to obtain the unknown coefficients and the approximate solution of the integral equation.