Discretization of Operator Equations

Many mathematical models are derived in abstract, and often infinite-dimensional, spaces. Frequently, the resulting equations are operator equations which have to be solved numerically. This implies that one has to perform finitization processes yielding operator equations over finite dimensional spaces which can be solved numerically. In the present chapter we study how these approximative solutions converge to an approximation of the solution of the original problem. First, we derive methods that allow for discrete approximations of metric spaces and of operators defined over such spaces. Second, we describe the concept of collectively compact operator approximations and investigate its fundamental properties. Finally, we apply the theory developed so far to the problem of how to solve numerically Fredholm integral equations of the second kind.