Adaptive Error Control for Multigrid Finite Element Methods (original) (raw)
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Contemporary Mathematics, 1994
The fully adaptive multigrid method (FAMe) is a nite element based elliptic solver integrating self-adaptivity, error estimation and e cient iterative solution. Re ned elements are not restricted to predetermined regions and need not be grouped in patches. Instead, whether an element is re ned, is decided individually for each element using an integrated error indicator. The re nement process induces a multilevel structure and therefore a natural decomposition of the solution space into a nested sequence. This can be exploited to de ne an e cient solver and error estimator.
Introduction to Multigrid Methods for Elliptic Boundary Value Problems
We treat multigrid methods for the efficient iterative solution of discretized elliptic boundary value problems. Two model problems are the Poisson equation and the Stokes problem. For the discretization we use standard finite element spaces. After discretization one obtains a large sparse linear system of equations. We explain multigrid methods for the solution of these linear systems. The basic concepts underlying multigrid solvers are discussed. Results of numerical experiments are presented which demonstrate the efficiency of these method. Theoretical convergence analyses are given that prove the typical grid independent convergence of multigrid methods.
2006
Abstract—We are concerned with an a posteriori error analysis of adaptive finite element approximations of boundary control problems for second order elliptic boundary value problems under bilateral bound constraints on the control which acts through a Neumann type boundary condition. In particular, the analysis of the errors in the state, the co-state, the control, and the co-control invokes an efficient and reliable residual-type a posteriori error estimator as well as data oscillations.
On Multigrid Convergence for Quadratic Finite Elements
2000
Quadratic and higher order finite elements are interesting candidates for the numerical solution of (elliptic) partial differential equations (PDEs) due to their improved approx- imation properties in comparison to linear approaches. While the systems of equations that arise from the discretisation of the underlying PDEs are often solved by iterative schemes like preconditioned Krylow-space methods, multigrid solvers are still rarely
Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients
2012
This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-pAMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigenproblems (corresponding to eigenvalues close to zero) form the coarse basis used to define the new two-level method of our interest. We illustrate the performance of this adaptive two-level procedure with a large set of numerical experiments that demonstrate its efficiency over building the solvers from scratch.
A Multigrid Method for Nonlinear Unstructured Finite Element Elliptic Equations
Siam Journal on Scientific Computing
Abstract: This paper presents an application of the element agglomeration-based coarsening procedure (agglomeration AMGe) proposed in [10], to build the components of a multigrid method for solving nonlinear finite element elliptic equations on general unstructured meshes. The agglomeration-based AMGe offers the ability to define coarse elements and element matrices, provided access to elements and element matrices on the fine grid is available. We focus on the performance of the classical full...
A posteriori error control for finite element approximations of elliptic eigenvalue problems
2001
We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.
The cascadic multigrid method for elliptic problems
Numerische Mathematik, 1996
The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of . From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on finer grids. A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, which used the CG method as basic iteration method on each level. In it has been proven, that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasi-uniform triangulations. The present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non-uniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and robustness of the cascadic multigrid method.
Adaptive refinement criterion for elliptic problems discretized by FEM
Communications in Numerical Methods in Engineering, 1993
In a recent paper' we presented a data structure to be used with multigrid techniques on nonhomogeneously refined FEM meshes. This paper focuses on the adaptive refinement techniques used there. The error estimate is obtained from standard Taylor series. For each element we compute its efficiency in terms of the size, the norm of the second derivatives of the unknown and the parameter p,