Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential ...Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.展开更多
In this paper,we design an efficient domain decomposition(DD)preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems.By proper equivalent alge...In this paper,we design an efficient domain decomposition(DD)preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems.By proper equivalent algebraic operations,the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonel matrix efficiently.Actually,the first block of this block-diagonal matrix corresponds to a multiscale H(div)problem,and thus,the direct inverse of this block is unpractical and unstable for the large-scale problem.To remedy this issue,a two-level overlapping DD preconditioner is proposed for this//(div)problem.Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis(e.g.,Raviart-Thomas element)on the coarse grid.The condition number of our preconditioned DD method for this multiscale H(div)system is bounded by C(1+务)(1+log4(^)),where 6 denotes the width of overlapping region,and H,h are the typical sizes of the subdomain and fine mesh.Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.展开更多
文摘Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.
文摘In this paper,we design an efficient domain decomposition(DD)preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems.By proper equivalent algebraic operations,the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonel matrix efficiently.Actually,the first block of this block-diagonal matrix corresponds to a multiscale H(div)problem,and thus,the direct inverse of this block is unpractical and unstable for the large-scale problem.To remedy this issue,a two-level overlapping DD preconditioner is proposed for this//(div)problem.Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis(e.g.,Raviart-Thomas element)on the coarse grid.The condition number of our preconditioned DD method for this multiscale H(div)system is bounded by C(1+务)(1+log4(^)),where 6 denotes the width of overlapping region,and H,h are the typical sizes of the subdomain and fine mesh.Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.
