On some boundary element methods for the heat equation
The Dirichlet problem and the Neumann problem for the heat equation in one space dimension with given initial values are considered. In the first approach the boundary integral equation based on the heat equation is approximately solved by a piecewise constant ansatz for the normal derivative in the Dirichlet problem, by a piecewise linear ansatz for the boundary values of the solution of the differential equation in the Neumann problem (discretization by a constant time-step). The approximate solution is then obtained by aid of the integral representation. In the second approach the boundary element technique is applied to the set of elliptic problems arising from discretization of the heat equation in time. The convergence analysis is carried out for both approaches.
scientific article; zbMATH DE number 3936433