Monotone iterates for solving coupled systems of nonlinear parabolic equations
When finite difference or finite element methods are applied to solve numerically systems of nonlinear parabolic partial differential equations, the corresponding discrete problem on each discrete time level is usually formulated as a system of nonlinear algebraic equations. A basic issue then is to guarantee that this system has a solution. Typically, some iterative method is used to construct numerical approximations to the resulting algebraic equations, so that the question of the convergence of the sequence of iterations is also relevant in this setting. In the paper under review, the technique of upper and lower solutions is considered as the iterative scheme. By using upper and lower solutions as initial iterations, two monotone sequences are constructed that converge monotonically from above and below to a solution of the problem. An analysis of convergence rates is also provided. In addition, a monotone domain decomposition for systems of nonlinear parabolic equations is achieved by combining the monotone approach with an iterative domain decomposition method. A numerical example derived from a gas-liquid interaction model is considered for illustrating the main theoretical results.
systems of nonlinear parabolic equations
scientific article; zbMATH DE number 5968772