A defect-correction method for unsteady conduction-convection problems II: Time discretization

In this paper, a fully discrete defect-correction mixed finite element method (MFEM) for solving the non-stationary conduction-convection problems in two dimension, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect-correction technique. The stability and the error analysis are derived. The theory analysis shows that our method is stable and has a good convergence property. Some numerical results are also given, which show that this method is highly efficient for the unsteady conduction-convection problems.     

Error estimate of a fully discrete defect correction finite element method for unsteady incompressible Magnetohydrodynamics equations

In this study, a fully discrete defect correction finite element method for the unsteady incompressible Magnetohydrodynamics (MHD) equations, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. It is a continuous work of our formal paper [Math Method Appl Sci. 2017. DOI:10.1002/mma.4296]. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear MHD equation is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect-correction technique. Then, we introduce the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. Some numerical results [see Math Method Appl Sci. 2017. DOI:10.1002/mma.4296] show that this method is highly efficient for the unsteady incompressible MHD problems.     

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.     

Use of Shifted Laplacian Operators for Solving Indefinite Helmholtz Equations

A shifted Laplacian operator is obtained from the Helmholtz operator by adding a complex damping. It serves as a basic tool in the most successful multigrid approach for solving highly indefinite Helmholtz equations — a Shifted Laplacian preconditioner for Krylov-type methods. Such preconditioning significantly accelerates Krylov iterations, much more so than the multigrid based on original Helmholtz equations. In this paper, we compare approximation and relaxation properties of the Helmholtz operator with and without the complex shift, and, based on our observations, propose a new hybrid approach that combines the two. Our analytical conclusions are supported by two-dimensional numerical results.     

Local Fourier analysis for multigrid with overlapping smoothers applied to systems of PDEs

Since their popularization in the late 1970s and early 1980s, multigrid methods have been a central tool in the numerical solution of the linear and nonlinear systems that arise from the discretization of many PDEs. In this paper, we present a local Fourier analysis (LFA, or local mode analysis) framework for analyzing the complementarity between relaxation and coarse‐grid correction within multigrid solvers for systems of PDEs. Important features of this analysis framework include the treatment of arbitrary finite‐element approximation subspaces, leading to discretizations with staggered grids, and overlapping multiplicative Schwarz smoothers. The resulting tools are demonstrated for the Stokes, curl–curl, and grad–div equations. Copyright © 2011 John Wiley & Sons, Ltd.     

Rate of Convergence for some constraint decomposition methods for nonlinear variational inequalities

Summary. Some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset. One of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods. For domain decomposition and multigrid methods, the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems. In order to analyse the convergence rate, we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints. A special nonlinear interpolation operator is introduced for decomposing the functions. Received December 13, 2001 / Revised version received February 19, 2002 / Published online June 17, 2002 This work was partially supported by the Norwegian Research Council under projects 128224/431 and SEP-115837/431.     

A defect-correction method for unsteady conduction convection problems Ⅰ:spatial discretization

In this paper, a semi-discrete defect-correction mixed finite element method (MFEM) for solving the non-stationary conduction-convection problems in two dimension is presented. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct this solutions on the same grid using a linearized defect-correction technique. The stability and the error analysis are derived. The theory analysis shows that our method is stable and has a good convergence ...     

A convergent multigrid cycle for the hybridized mixed method

We consider the application of a variable V‐cycle multigrid algorithm for the hybridized mixed method for second‐order elliptic boundary‐value problems. Our algorithm differs from the previous works on multigrid for the mixed method in that it is targeted at efficiently solving the matrix system for the Lagrange multiplier of the method. Since the mixed method is best implemented by first solving for the Lagrange multiplier and recovering the remaining unknowns locally, our algorithm is more useful in practice. The critical ingredient in the algorithm is a suitable intergrid transfer operator. We design such an operator and prove mesh‐independent convergence of the variable V‐cycle algorithm. Numerical experiments indicating the asymptotically optimal performance of our algorithm, as well as the failure of certain seemingly plausible intergrid transfer operators, are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Multigrid algorithms for a vertex–centered covolume method for elliptic problems

Summary. We analyze V–cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem. As an application, we study the convergence of multigrid algorithms for a covolume method or a vertex–centered finite volume element method for variable coefficient elliptic problems on polygonal domains. As in standard finite element methods, the V–cycle algorithm with one pre-smoothing converges with a rate independent of the number of levels. Various types of smoothers including point or line Jacobi, and Gauss-Seidel relaxation are considered. Received August 19, 1999 / Revised version received July 10, 2000 / Published online June 7, 2001     

A Vanka-based parameter-robust multigrid relaxation for the Stokes–Darcy Brinkman problems

We consider a block-structured multigrid method based on Braess–Sarazin relaxation for solving the Stokes–Darcy Brinkman equations discretized by the marker and cell scheme. In the relaxation scheme, an element-based additive Vanka operator is used to approximate the inverse of the corresponding shifted Laplacian operator involved in the discrete Stokes–Darcy Brinkman system. Using local Fourier analysis, we present the stencil for the additive Vanka smoother and derive an optimal smoothing factor for Vanka-based Braess–Sarazin relaxation for the Stokes–Darcy Brinkman equations. Although the optimal damping parameter is dependent on meshsize and physical parameter, it is very close to one. In practice, we find that using three sweeps of Jacobi relaxation on the Schur complement system is sufficient. Numerical results of two-grid and V(1,1)-cycle are presented, which show high efficiency of the proposed relaxation scheme and its robustness to physical parameters and the meshsize. Using a damping parameter equal to one gives almost the same convergence results as these for the optimal damping parameter.     

A new level‐dependent coarse grid correction scheme for indefinite Helmholtz problems

In this paper, we construct and analyze a level‐dependent coarse grid correction scheme for indefinite Helmholtz problems. This adapted multigrid (MG) method is capable of solving the Helmholtz equation on the finest grid using a series of MG cycles with a grid‐dependent complex shift, leading to a stable correction scheme on all levels. It is rigorously shown that the adaptation of the complex shift throughout the MG cycle maintains the functionality of the two‐grid correction scheme, as no smooth modes are amplified in or added to the error. In addition, a sufficiently smoothing relaxation scheme should be applied to ensure damping of the oscillatory error components. Numerical experiments on various benchmark problems show the method to be competitive with or even outperform the current state‐of‐the‐art MG‐preconditioned Krylov methods, for example, complex shifted Laplacian preconditioned flexible GMRES. Copyright © 2013 John Wiley & Sons, Ltd.     

A multilevel finite volume method with multiscale-based grid adaptation for steady compressible flows

An implicit multilevel finite volume solver on adaptively refined quadtree meshes is presented for the solution of steady state flow problems. The nonlinear problem arising from the implicit time discretization is solved by an adaptive FAS multigrid method. Local grid adaptation is performed by means of a multiscale-based strategy. For this purpose data of the flow field are decomposed into coarse grid information and a sequence of detail coefficients that describe the difference between two refinement levels and reveal insight into the local regularity behavior of the solution. Here wavelet techniques are employed for the multiscale analysis. The key idea of the present work is to use the transfer operators of the multiscale analysis for the prolongation and restriction operator in the FAS cycle. The efficiency of the solver is investigated by means of an inviscid 2D flow over a bump.     

A multigrid-based preconditioned Krylov subspace method for the Helmholtz equation with PML

In this paper, we generalize the complex shifted Laplacian preconditioner to the complex shifted Laplacian-PML preconditioner for the Helmholtz equation with perfectly matched layer (Helmholtz-PML equation). The Helmholtz-PML equation is discretized by an optimal 9-point difference scheme, and the preconditioned linear system is solved by the Krylov subspace method, especially by the biconjugate gradient stabilized method (Bi-CGSTAB). The spectral analysis of the linear system is given, and a new matrix-based interpolation operator is proposed for the multigrid method, which is used to approximately invert the preconditioner. The numerical experiments are presented to illustrate the efficiency of the preconditioned Bi-CGSTAB method with the multigrid based on the new interpolation operator, also, numerical results are given for comparing the performance of the new interpolation operator with that of classic bilinear interpolation operator and the one suggested in Erlangga et al. (2006) [10].     

Convergence rate analysis of an asynchronous space decomposition method for convex minimization

We analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive Banach space. This method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations. In particular, the method generalizes the additive Schwarz domain decomposition methods to allow for asynchronous updates. It also generalizes the BPX multigrid method to allow for use as solvers instead of as preconditioners, possibly with asynchronous updates, and is applicable to nonlinear problems. Applications to an overlapping domain decomposition for obstacle problems are also studied. The method of this work is also closely related to relaxation methods for nonlinear network flow. Accordingly, we specialize our convergence rate results to the above methods. The asynchronous method is implementable in a multiprocessor system, allowing for communication and computation delays among the processors.

     

Box-Line Relaxation Schemes for Solving the Steady Incompressible Navier-Stokes Equations Using Second-Order Upwind Differencing

1.Introducti0nThec0nvection-diffusionbehaviour0fthevisc0usincompressibleNavier-St0kesequati0nsisamains0urce0fdifficultiesinthenumericalsoluti0n.Whendiscretiz-ingtheequati0nsusingfinitedifferenceschemes,upwind0rhybridschemesareusuallyused0nthec0nvectiontermsforensuringthestability0fthediscretesystem[1].Thefirst-orderuPwinddifferencinghasprovedt0beinadequatef0rtheincompressibleNavier-St0kesequati0nswithlargeReynoldsnumbers,alth0ughtheresultingdiscretesystemsareverystableandeasilys0lved.In[5],…     

Comparison of multigrid algorithms for high‐order continuous finite element discretizations

We present a comparison of different multigrid approaches for the solution of systems arising from high‐order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the pointwise Jacobi, the Chebyshev‐accelerated Jacobi, and the symmetric successive over‐relaxation smoothers, as well as elementwise block Jacobi smoothing. Three approaches for the multigrid hierarchy are compared: (1) high‐order h‐multigrid, which uses high‐order interpolation and restriction between geometrically coarsened meshes; (2) p‐multigrid, in which the polynomial order is reduced while the mesh remains unchanged, and the interpolation and restriction incorporate the different‐order basis functions; and (3) a first‐order approximation multigrid preconditioner constructed using the nodes of the high‐order discretization. This latter approach is often combined with algebraic multigrid for the low‐order operator and is attractive for high‐order discretizations on unstructured meshes, where geometric coarsening is difficult. Based on a simple performance model, we compare the computational cost of the different approaches. Using scalar test problems in two and three dimensions with constant and varying coefficients, we compare the performance of the different multigrid approaches for polynomial orders up to 16. Overall, both h‐multigrid and p‐multigrid work well; the first‐order approximation is less efficient. For constant coefficients, all smoothers work well. For variable coefficients, Chebyshev and symmetric successive over‐relaxation smoothing outperform Jacobi smoothing. While all of the tested methods converge in a mesh‐independent number of iterations, none of them behaves completely independent of the polynomial order. When multigrid is used as a preconditioner in a Krylov method, the iteration number decreases significantly compared with using multigrid as a solver. Copyright © 2015 John Wiley & Sons, Ltd.     

A new subspace correction method for nonlinear unconstrained convex optimization problems

This paper gives a new subspace correction algorithm for nonlinear unconstrained convex optimization problems based on the multigrid approach proposed by S. Nash in 2000 and the subspace correction algorithm proposed by X. Tai and J. Xu in 2001. Under some reasonable assumptions, we obtain the convergence as well as a convergence rate estimate for the algorithm. Numerical results show that the algorithm is effective.     

A note on an accelerated high‐accuracy multigrid solution of the convection‐diffusion equation with high Reynolds number

We present a new strategy to accelerate the convergence rate of a high‐accuracy multigrid method for the numerical solution of the convection‐diffusion equation at the high Reynolds number limit. We propose a scaled residual injection operator with a scaling factor proportional to the magnitude of the convection coefficients, an alternating line Gauss‐Seidel relaxation, and a minimal residual smoothing acceleration technique for the multigrid solution method. The new implementation strategy is tested to show an improved convergence rate with three problems, including one with a stagnation point in the computational domain. The effect of residual scaling and the algebraic properties of the coefficient matrix arising from the fourth‐order compact discretization are investigated numerically. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 1–10, 2000     

Weighted relaxation for multigrid reduction in time

Current trends in computer architectures now mean that faster computation speed must come primarily from increased concurrency, not faster clock speeds, which are stagnating. Thus, this situation creates bottlenecks for serial algorithms, including the well-known bottleneck for sequential time-integration, where each individual time-value (i.e., time-step) is computed sequentially. One approach to alleviate this and achieve parallelism in time is with multigrid. In this work, we consider multigrid-reduction-in-time (MGRIT), a multilevel method applied to the time dimension that computes multiple time-steps in parallel. Like all multigrid methods, MGRIT relies on the complementary relationship between relaxation on a fine-grid and a correction from the coarse grid to solve the problem. All current MGRIT implementations are based on unweighted-Jacobi relaxation; here we introduce the concept of weighted relaxation to MGRIT. We derive new convergence bounds for weighted relaxation, and use this analysis to guide the selection of relaxation weights. Numerical results then demonstrate that by choosing appropriate non-unitary relaxation weights, one can achieve faster convergence rates and lower iteration counts for MGRIT when compared with unweighted relaxation. In most cases, weighted relaxation yields a 10%–20% saving in iterations, which is significant when using large high-performance computers. For A-stable integration schemes, results also illustrate that under-relaxation can restore convergence in some cases where unweighted relaxation is not convergent.