An auxiliary space multigrid preconditioner for the weak Galerkin method (original) (raw)
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Preconditioning and Two-Level Multigrid Methods of Arbitrary Degree of Approximation
Mathematics of Computation, 1983
Let A be a mesh parameter corresponding to a finite element mesh for an elliptic problem. We describe preconditioning methods for two-level meshes which, for most problems solved in practice, behave as methods of optimal order in both storage and computational complexity. Namely, per mesh point, these numbers are bounded above by relatively small constants for all h > h0, where h0 is small enough to cover all but excessively fine meshes. We note that, in practice, multigrid methods are actually solved on a finite, often even a fixed number of grid levels, in which case also these methods are not asymptotically optimal as h-> 0. Numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems (variable coefficients, general domains and boundary conditions) for all but excessively fine meshes. Furthermore, most of the latter methods have been implemented only for difference schemes of second order of accuracy, whereas our methods are applicable to higher order approximations. We claim that our scheme could be added fairly easily to many existing finite element codes.
Efficient preconditioning for the discontinuous Galerkin finite element method by low-order elements
Applied Numerical Mathematics, 2009
We derive and analyze a block diagonal preconditioner for the linear problems arising from a discontinuous Galerkin finite element discretization. The method can be applied to second-order self-adjoint elliptic boundary value problems and exploits the natural decomposition of the discrete function space into a global low-order subsystem and components of higher polynomial degree. Similar to results for the p-version of the conforming FEM, we prove for the interior penalty discontinuous Galerkin discretization that the condition number of the preconditioned system is uniformly bounded with respect to the mesh size of the triangulation. Numerical experiments demonstrate the performance of the method.
Uniciencia, 2020
This paper presents detailed aspects regarding the implementation of the Finite Element Method (FEM) to solve a Poisson’s equation with homogeneous boundary conditions. The aim of this paper is to clarify details of this implementation, such as the construction of algorithms, implementation of numerical experiments, and their results. For such purpose, the continuous problem is described, and a classical FEM approach is used to solve it. In addition, a multilevel technique is implemented for an efficient resolution of the corresponding linear system, describing and including some diagrams to explain the process and presenting the implementation codes in MATLAB®. Finally, codes are validated using several numerical experiments. Results show an adequate behavior of the preconditioner since the number of iterations of the PCG method does not increase, even when the mesh size is reduced.
2013
This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral DG discretizations of elliptic boundary value problems. The far term goal is to obtain robust solvers for the "fully flexible" case. By this we mean Discontinuous Galerkin schemes on locally refined quadrilateral or hexahedral partitions with hanging nodes and variable polynomial degrees that could, in principle, be arbitrarily large only subject to some weak grading constraints. In this paper, as a first step, we focus on varying arbitrarily large degrees while keeping the mesh geometrically conforming since this will be seen to exhibit already some essential obstructions. The conceptual foundation of the envisaged preconditioners is the auxiliary space method, or in fact, an iterated variant of it. The main conceptual pillars that will be shown in this framework to yield "optimal" preconditioners are Legendre-Gauß-Lobatto grids in connection with certain associated anisotropic nested dyadic grids. Here "optimal" means that the preconditioned systems exhibit uniformly bounded condition numbers. Moreover, the preconditioners have a modular form that facilitates somewhat simplified partial realizations at the expense of a moderate loss of efficiency. Our analysis is complemented by careful quantitative experimental studies of the main components.
MultiGrid Preconditioners for Mixed Finite Element Methods of the Vector Laplacian
Journal of Scientific Computing, 2018
Due to the indefiniteness and poor spectral properties, the discretized linear algebraic system of the vector Laplacian by mixed finite element methods is hard to solve. A block diagonal preconditioner has been developed and shown to be an effective preconditioner by Arnold et al. (Acta Numer 15:1–155, 2006). The purpose of this paper is to propose alternative and effective block diagonal and approximate block factorization preconditioners for solving these saddle point systems. A variable V-cycle multigrid method with the standard point-wise Gauss–Seidel smoother is proved to be a good preconditioner for the discrete vector Laplacian operator. The major benefit of our approach is that the point-wise Gauss–Seidel smoother is more algebraic and can be easily implemented as a black-box smoother. This multigrid solver will be further used to build preconditioners for the saddle point systems of the vector Laplacian. Furthermore it is shown that Maxwell’s equations with the divergent fr...
Numerische Mathematik, 2015
In this paper a discretization based on discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region Ω which is a union of N disjoint polygonal subdomains Ω i of diameter O(H i). The discontinuities of the coefficients, possibly very large, are assumed to occur only across the subdomain interfaces ∂Ω i. In each Ω i a conforming quasiuniform triangulation with parameters h i is constructed. We assume that the resulting triangulation in Ω is also conforming, i.e., the meshes are assumed to match across the subdomain interfaces. On the fine triangulation the problem is discretized by a DG method. For solving the resulting discrete system, a FETI-DP type method is proposed and analyzed. It is established that the condition number of the preconditioned linear system is estimated by C(1 + max i log H i /h i) 2 with a constant C independent of h i , H i and the jumps of coefficients. The method is well suited for parallel computations and it can be extended to threedimensional problems. This result is an extension, to the case of full fine-grid DG discretization, of the previous result [SIAM J. Numer. Anal., 51 (2013), pp. 400-422] where it was considered a conforming finite element method
Multigrid preconditioning in H(div) on non-convex polygons
1998
In an earlier paper we constructed and analyzed a multigrid preconditioner for the system of linear algebraic equations arising from the finite element discretization of boundary value problems associated to the differential operator I − grad div. In this paper we analyze the procedure without assuming that the underlying domain is convex and show that, also in this case, the preconditioner is spectrally equivalent to the inverse of the discrete operator.
The analysis of FETI-DP preconditioner for full DG discretization of elliptic problems
2014
In this paper a discretization based on discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region Omega\OmegaOmega which is a union of NNN disjoint polygonal subdomains Omegai\Omega_iOmegai of diameter O(Hi)O(H_i)O(Hi). The discontinuities of the coefficients, possibly very large, are assumed to occur only across the subdomain interfaces partialOmegai\partial \Omega_ipartialOmegai. In each Omegai\Omega_iOmegai a conforming quasiuniform triangulation with parameters hih_ihi is constructed. We assume that the resulting triangulation in Omega\OmegaOmega is also conforming, i.e., the meshes are assumed to match across the subdomain interfaces. On the fine triangulation the problem is discretized by a DG method. For solving the resulting discrete system, a FETI-DP type method is proposed and analyzed. It is established that the condition number of the preconditioned linear system is estimated by C(1+maxilogHi/hi)2C(1 + \max_i \log H_i/h_i)^2C(1+maxilogHi/hi)2 with a constant CCC independent of hih_ihi, HiH_iHi and the jumps of coefficients. The method is well suited for parallel computations and it can be extended to three-dimensional problems. This result is an extension, to the case of full fine-grid DG discretization, of the previous result [SIAM J. Numer. Anal., 51 (2013), pp.~400--422] where it was considered a conforming finite element method inside the subdomains and a discontinuous Galerkin method only across the subdomain interfaces. Numerical results are presented to validate the theory.