Numerical methods and existence theorems for parabolic differential equations whose coefficients are singular on the boundary

I. Introduction.In a previous paper [6], S. V. Parter and the author have studied finite-difference methods for elliptic differential equations of the second order whose coefficients are singular on a portion of the boundary; the uniform convergence of the approximations and the existence of a solution of the Dirichlet problem were proved for a class of such equations.The present work is an extension of those results to parabolic initial boundary-value problems.The class of problems that we consider includes the cases of nonhomogeneous differential equations, of time-dependent coefficients, of time-dependent domains and of over-determined Dirichlet problems.Let G be a bounded (open) domain in Rn and let P = (xi, • • •, xn) denote an element of G. Let L be a differential operator of the form