The Research On Newton-Krylov-Schwarz Algorithm For The Microwave Heating Equation

Partial differential equations(PDEs)can be applied in the description of various phenomena in engineering,physics,economics,and biology.These partial differential equations are usually very complicated and have no analytical solution,so we can only resort to numerical methods to obtain the numerical solution.Therefore,searching for a high-performance and robust algorithm for the PDEs has become a quite popular topic in the field of scientific computing.After discretizing the partial differential equation we obtain a linear or nonlinear algebraic system.Hence,this thesis focuses on the high-performance algorithm for solving a nonlinear system F(x)= 0(F:Rn?Rn).This thesis initially introduces the family of Newton methods for solving non-linear equation systems and the Krylov subspace iterative technique for solving the corresponding linear systems.Then Newton-Krylov method is the combination of the Newton algorithm and the Krylov subspace technique.Since there are many steps of linear iteration in each Newton iteration process.In the process of the linear iteration,a Jacobian linear system need to be solved.The corresponding Jacobian matrix is com-monly ill-conditioned,large-scale sparse and asymmetric.Solving this linear system directly consumes much computational memory resources and is inefficient and even impossible.For these reasons,we use the preconditioning technique to accelerate the process of solving the linear systems.The construction and selection of preconditioner also determine the effect of the computation.It involves the domain decomposition method.This thesis uses the Schwarz preconditioner and combines it with the above methods to get the Newton-Krylov-Schwarz algorithm.Secondly,for a class of varia-tional inequality problems,which has wide applications in engineering,the author also proposes a semismooth Newton-Krylov-Schwarz algorithm.This method uses a nonlin-ear complementary function to transform the variational inequality problem into a class of semismooth nonlinear equations problems.Finally,for a class of unsteady problems,the author proposes a fully implicit Newton-Krylov-Schwarz algorithm,which has the advantage of allowing large time steps.For the above mentioned algorithms,a series of numerical experiments are performed in this thesis.The numerical results show that proposed algorithms are effective.