AN A.D.I.GALERKIN METHOD FOR NONLINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATION USING PATCH APPROXIMATION
摘要
An A. D. I. Galerkin scheme for three-dimensional nonlinear parabolic integro-differen-tial equation is studied. By using alternating-direction, the three-dimensional problem is reduced to a family of single space variable problems, the calculation is simplified; by using a local approxima-tion of the coefficients based on patches of finite elements, the coefficient matrix is updated at each time step; by using Ritz-Volterra projection, integration by part and other techniques, the influence coming from the integral term is treated; by using inductive hypothesis reasoning, the difficulty coming from the nonlinearity is treated. For both Galerkin and A. D. I. Galerkin schemes the con-vergence properties are rigorously demonstrated, the optimal H~1-norm and L~2-norm estimates are obtained.