Design of Finite Element Tools for Coupled Surface and Volume Meshes

Many problems with underlying variational structure involve a coupling of volume with surface effects.A straight-forward approach in a finite element discretiza- tion is to make use of the surface triangulation that is naturally induced by the volume triangulation.In an adaptive method one wants to facilitate"matching"local mesh modifications,i.e.,local refinement and/or coarsening,of volume and surface mesh with standard tools such that the surface grid is always induced by the volume grid. We describe the concepts behind this approach for bisectional refinement and describe new tools incorporated in the finite element toolbox ALBERTA.We also present several important applications of the mesh coupling.     

MULTIGRID FOR THE MORTAR ELEMENT METHOD WITH LOCALLY P_1 NONCONFORMING ELEMENTS

In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain are uniform. We prove the convergence of the W-cycle with a sufficiently large number of smoothing steps. The     

Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems

In this paper we prove the uniform convergence of the standard multigrid V-cycle algorithm with the Gauss-Seidel relaxation performed only on the new nodes and their "immediate" neighbors for discrete elliptic problems on the adaptively refined finite element meshes using the newest vertex bisection algorithm. The proof depends on sharp estimates on the relationship of local mesh sizes and a new stability estimate for the space decomposition based on the Scott-Zhang interpolation operator. Extensive numerical results are reported, which confirm the theoretical analysis.     

MONOLITHIC MULTIGRID FOR REDUCED MAGNETOHYDRODYNAMIC EQUATIONS

In this paper,the monolithic multigrid method is investigated for reduced magnetohydrodynamic equations.We propose a diagonal Braess-Sarazin smoother for the finite element discrete system and prove the uniform convergence of the MMG method with respect to mesh sizes.A multigrid-preconditioned FGMRES method is proposed to solve the magnetohydrodynamic equations.It turns out to be robust for relatively large physical parameters.By extensive numerical experiments,we demonstrate the optimality of the monolithic multigrid method with respect to the number of degrees of freedom.     

HANGING NODES IN THE UNIFYING THEORY OF A POSTERIORI FINITE ELEMENT ERROR CONTROL

A unified a posteriori error analysis has been developed in [18, 21-23] to analyze the finite element error a posteriori under a universal roof. This paper contributes to the finite element meshes with hanging nodes which are required for local mesh-refining. The twodimensional 1-irregular triangulations into triangles and parallelograms and their combinations are considered with conforming and nonconforming finite element methods named after or by Courant, Q1, Crouzeix-Raviart, Poisson, Stokes and Navier-Lamé equations Han, Rannacher-Turek, and others for the The paper provides a unified a priori and a posteriori error analysis for triangulations with hanging nodes of degree ≤ 1 which are fundamental for local mesh refinement in self-adaptive finite element discretisations.     

Hybrid Surface Mesh Adaptation for Climate Modeling

Solution-driven mesh adaptation is becoming quite popular for spatial error control in the numerical simulation of complex computational physics applications, such as climate modeling. Typically, spatial adaptation is achieved by element subdivision （h adaptation） with a primary goal of resolving the local length scales of interest. A sec- ond, less-popular method of spatial adaptivity is called ＂mesh motion＂ （r adaptation）; the smooth repositioning of mesh node points aimed at resizing existing elements to capture the local length scales. This paper proposes an adaptation method based on a combination of both element subdivision and node point repositioning （rh adaptation）. By combining these two methods using the notion of a mobility function, the proposed approach seeks to increase the flexibility and extensibility of mesh motion algorithms while providing a somewhat smoother transition between refined regions than is pro- duced by element subdivision alone. Further, in an attempt to support the requirements of a very general class of climate simulation applications, the proposed method is designed to accommodate unstructured, polygonal mesh topologies in addition to the most popular mesh types.     

ADAPTIVITY IN SPACE AND TIME FOR MAGNETOQUASISTATICS

This paper addresses fully space-time adaptive magnetic field computations. We describe an adaptive Whitney finite element method for solving the magnetoquasistatic formulation of Maxwell＇s equations on unstructured 3D tetrahedral grids. Spatial mesh re- finement and coarsening are based on hierarchical error estimators especially designed for combining tetrahedral H（curl）-conforming edge elements in space with linearly implicit Rosenbrock methods in time. An embedding technique is applied to get efficiency in time through variable time steps. Finally, we present numerical results for the magnetic recording write head benchmark problem proposed by the Storage Research Consortium in Japan.     

ESTIMATOR COMPETITION FOR POISSON PROBLEMS

We compare 13 different a posteriori error estimators for the Poisson problem with lowest-order finite element discretization. Residual-based error estimators compete with a wide range of averaging estimators and estimators based on local problems. Among our five benchmark problems we also look on two examples with discontinuous isotropic diffusion and their impact on the performance of the estimators. （Supported by DFG Research Center MATHEON.）     

A second order isoparametric finite element method for elliptic interface problems

A second order isoparametric finite element method (IPFEM) is proposed for elliptic interface problems. It yields better accuracy than some existing second-order methods, when the coefficients or the flux across the immersed curved interface is discontinuous. Based on an initial Cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at the interface, and an incomplete quadratic finite element space is constructed on the optimized mesh. It turns out that the number of curved boundary elements is far less than that of the straight one, and the total degree of freedom is almost the same as the uniform Cartesian mesh. Numerical examples with simple and complicated geometrical interfaces demonstrate the efficiency of the proposed method.     

A POSTERIORI ERROR ESTIMATE FOR BOUNDARY CONTROL PROBLEMS GOVERNED BY THE PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we discuss the a posteriori error estimate of the finite element approximation for the boundary control problems governed by the parabolic partial differential equations. Three different a posteriori error estimators are provided for the parabolic boundary control problems with the observations of the distributed state, the boundary state and the final state. It is proven that these estimators are reliable bounds of the finite element approximation errors, which can be used as the indicators of the mesh refinement in adaptive finite element methods.     

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate" neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.     

Uniform Convergence of Multigrid V-Cycle on Adaptively Refined Finite Element Meshes for Elliptic Problems with Discontinuous Coefficients

The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered. Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm, some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours. The multigrid V-cycle algorithm uses $\mathcal{O}(N)$ operations per iteration and is optimal.     

Convergence Estimates of Multilevel Additive and Multiplicative Algorithms for Non-symmetric and Indefinite Problems

New uniform estimates for multigrid algorithms are established for certain non-symmetric indefinite problems. In particular, we are concerned with the simple additive algorithm and multigrid (V(1,0)-cycle) algorithms given in [5]. We prove, without full elliptic regularity assumption, that these algorithms have uniform reduction per iteration, independent of the finest mesh size and number of refinement levels, provided that the coarsest mesh size is sufficiently small.     

Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra

This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right‐hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi‐uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by‐product, finite element error estimates in the H¹(Ω)‐norm, L²(Ω)‐norm and L²(Γ)‐norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.     

On the Convergence of Adaptive Finite Element Methods

State of the art simulations in computational mechanics aim reliability and efficiency via adaptive finite element methods (AFEMs) with a posteriori error control. The a priori convergence of finite element methods is justified by the density property of the sequence of finite element spaces which essentially assumes a quasi‐uniform mesh‐refining. The advantage is guaranteed convergence for a large class of data and solutions; the disadvantage is a global mesh refinement everywhere accompanied by large computational costs. AFEMs automatically refine exclusively wherever the refinement indication suggests to do so and so violate the density property on purpose. Then, the a priori convergence of AFEMs is not guaranteed automatically and, in fact, crucially depends on algorithmic details. The advantage of AFEMs is a more effective mesh accompanied by smaller computational costs in many practical examples; the disadvantage is that the desirable error reduction property is not always guaranteed a priori. Efficient error estimators can justify a numerical approximation a posteriori and so achieve reliability. But it is not clear from the start that the adaptive mesh‐refinement will generate an accurate solution at all. This paper discusses particular versions of an AFEMs and their analyses for error reduction, energy reduction, and convergence results for linear and nonlinear problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error reduction and convergence for an adaptive mixed finite element method

An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart-Thomas finite element method with a reduction factor uniformly for the norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does not rely on duality or on regularity.

     

A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.     

Multigrid for the mortar element method for P1 nonconforming element

In this paper, a multigrid algorithm is presented for the mortar element method for P1 nonconforming element. Based on the theory developed by Bramble, Pasciak, Xu in [5], we prove that the W-cycle multigrid is optimal, i.e. the convergence rate is independent of the mesh size and mesh level. Meanwhile, a variable V-cycle multigrid preconditioner is constructed, which results in a preconditioned system with uniformly bounded condition number. Received May 11, 1999 / Revised version received April 1, 2000 / Published online October 16, 2000