ADFE method with high accuracy for nonlinear parabolic integro-differential system with nonlinear boundary conditions (Q1856772)

The author studies an alternating direction finite element (ADFE) scheme for a \(d\)-dimensional system of nonlinear parabolic integro-differential equations. The decomposition of the coefficient matrix is achieved by the use of a local approximation based on patches of finite elements (FE). The multi-dimensional problem is reduced to a family of single space variable problems by use of the alternating directions. A Ritz-Volterra projection is introduced to overcome the difficulty arising from the memory term. The unique resolvability and convergence properties for both FE and ADFE schemes are rigorously demonstrated by using various techniques for a priori estimates for differential equations.