The article considers a task of numerical solution of integral equations of electromagnetism, namely linear integral equations of second type, described on some multidimensional domain :
Here is unknown multidimensional function and K is a singular integral operator that acts in functional space . It is well-known fact that solution stability falls in case of bad task conditionality, but also convergence of iterative methods slows down essentially.
This article describes one way to solve the problem of very bad convergence of iterative methods. This problem is solved by equal transformation of source equation by means of left preconditioner operator :
Our approach for preconditioner building is based on the idea of inverse operator approximation on very coarse grid. The main problem at this point of view is requirement of such interpolation of retrieved approximation that would most naturally reflect source operator properties. For this problem solution we use transformations of source equation (1). Interpolation lies in incomplete data loss in coarse grid approximation.
By the way of equation (1) digitization we can build the linear algebraic equation system (LAES), described in finite-dimensional vector space . Dimension of vector space we consider is small enough, so that we can relatively easy invert the retrieved matrix of operator approximation. Let introduce operators between function space and vector space – and , such that satisfy the equality , where is identity operator in vector space . In this case the proposed precontitioner for iterative methods can be expressed by next formula:
Here is identity operator in functional space .
Considered an example of numerical solution of an electrodynamics task, which shows that, even for case of very bad equation conditionality, big difference of grids dimensions and too small coarse grid dimension for enough correct description of multidimensional singular equation, described preconditioner using piecewise constant operators and can essentially decrease number of iterations needed for solution by iterative methods.