Closed form numerical solutions of variable coefficient linear second-order elliptic problems

In this work we develop an alternative numerical technique which allows to construct a numerical solution in closed form of variable coefficient linear second-order elliptic problems with Dirichlet boundary conditions. The elliptic partial differential equation is approximated by a consistent explicit difference scheme and using a discrete separation of the variables method we determine a closed form solution of the two resulting discrete boundary value problems with the separated variables, avoiding to have to solve large algebraic systems. One of these boundary value problems is a discrete Sturm-Liouville problem which guarantees the qualitative properties of the exact solution of elliptic problem. A constructive procedure for the computation of the numerical solution is given and an illustrative example is included.