Efficient Multistep Procedures for Nonlinear Parabolic Problems with Nonlinear Neumann Boundary Conditions

摘要

Efficient multistep procedures for time-stepping Galerkin methods for nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions are presented and analyzed. The procedures involve using a preconditioned iterative method for approximately solving the different linear equations arising at each time step in a discrete time Galerkin method. Optimal order convergence rates are obtained for the iterative methods. Work estimates of almost optimal order are obtained. Many mathematical models for heat flow or fluid flow involve the specification of a flow rate across the boundary of a region which may depend in a nonlinear fashion upon the unknown variable (e.g. temperature). Formulation and analysis of efficient numerical procedures for approximating the solutions of such problems are studied. Previously, finite element methods used for modeling these physical problems were at most second order correct in the time-discretization error. Methods are produced which are second, third, and fourth order correct in time and which convert the nonlinear problems into solution of large systems of linear equations via an extremely stable algorithm with essentially no restrictions between sizes of time and space discretizations. The paper also contains work estimates which show the large computational savings of the preconditioned iterative stabilization technique. Almost optimal order work estimates are obtained.