Fast Algorithms Study Of High Order Finite Element Discretizations For Several Typical PDEs

H(D)(D=grad,curl,div)elliptic problems and Maxwell saddle-point problem are several classical partial differential equations(PDEs).High or-der finite element methods are the commonly used discretization techniques for solving them.But the condition number of the coefficient matrices cor-responding to the resulting discrete systems strongly depends on the mesh size,the jump coefficients and the order of the elements,etc,hence con-structing the corresponding fast algorithms is of great necessity.In this paper,using the auxiliary space preconditioner,algebraic multigrid(AMG)and nonoverlapping domain decomposition method(DDM),we discuss the efficient preconditioner and the corresponding fast algorithm for solving the resulting discrete systems of H(D)(D=grad,curl,div)elliptic problems and Maxwell saddle-point problem.The main contents of this dissertation are as follows.A parallel two-level preconditioner(TLB-p)based on the block Gauss-Seidel smooth method is given for the H(grad)elliptic problem with jump coefficients and solved by the high order element method with HBk hierachi-cal basis,it translate the construction of preconditioner for the high order element discretization into the one for the corresponding linear element dis-cretization.We gain the first preconditioner(TLB-AMG-p)for solving the discrete system of high order element by choosing BoomerAMG as the pre-conditioner of the linear element system.The numerical results indicate that the iteration number of the corresponding PCG algorithm is independent of the mesh size,weakly dependent on the order of basis function and the jump coefficients.Next,we design a nonoverlapping DDM preconditioner for the linear element discretization,it has some merits comparing with some exist-ing preconditioners,such as simple coarse spaces,low computing complica-tion and so on.The theoretical analysis and numerical experiments indicate that the effective condition number of the preconditioned system is nearly optimal.Then basing on this DDM preconditioner,we get another pre-conditioner(TLB-DDM-p)for high order finite element system.Numerical experiments also show the efficiency of the second preconditioner.By using the theory of the stable decompositions in high order edge element space and the above preconditioner for H(grad)elliptic problem,the construction of preconditioner for the discretization of the H(curl)el-liptic problem solved by the high order edge element method is translated to the one for the corresponding linear edge element discretization.Fur-thermore,we gain the first preconditioner for solving the high order edge element discrete system by choosing the auxiliary space preconditioner as the linear edge element system preconditioner.The numerical results indi-cate that the corresponding PCG algorithm has good algorithmic scalability,and the iteration number is independent of the mesh size,weakly dependent on the jump coefficients for no floating,floating,no cross point and cross point cases.Then,we design a nonoverlapping DDM preconditioner for the linear edge element discretization which has the same merits as the DDM preconditioner of H(grad)elliptic problem,and we prove,for the case with constant coefficients,that the condition number of the preconditioned sys-tem is nearly optimal.Numerical experiments confirm the theoretical results,and show that the preconditioner is also efficient for the case with large jump coefficients.Inspired by the idea of regularity,we design a new Uzawa algorithm based on the auxiliary space preconditioner for Maxwell saddle-point system with no zero order term which is discretized by the high order edge element.Especially,We prove that the convergence rate of the saddle-point system of linear edge element is independent of the mesh size if the coefficient is smooth.The numerical results indicate that the iteration number of the new Uzawa algorithm is nearly independent of the mesh size and the jump coefficients for no floating,floating,no cross point and cross point cases.Fur-thermore experiments show that the new algorithm owns better robustness and efficiency comparing with a commonly used Uzawa algorithm.We establish a theory of the stable decompositions for the first family and second family of high order divergence conforming element,then the construction of preconditioner for the high order element discretization is translated into the one for the corresponding linear element discretization.We obtain a preconditioner for high order finite element system by using the above preconditioner for H(curl)elliptic problem of high order edge element and an auxiliary space preconditioner for H(div)elliptic problem of linear element.We prove that the convergence rate is independent of the mesh size.Furthermore,the numerical results confirm the theoretical results and also show the efficiency and robustness of the new algorithm.