Graded Meshes for Higher Order FEM

A singularly perturbed one-dimensional convection-diffusion problem is solved numerically by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded (based on a mesh generating function) in the fine mesh region. Error estimates in the ε-weighted energy norm are proved. We derive an 'optimal' mesh generating function in order to minimize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving error estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results.     

An efficient hierarchical preconditioner for quadratic discretizations of finite element problems

Higher order finite element discretizations, although providing higher accuracy, are considered to be computationally expensive and of limited use for large‐scale problems. In this paper, we have developed an efficient iterative solver for solving large‐scale quadratic finite element problems. The proposed approach shares some common features with geometric multigrid methods but does not need structured grids to create the coarse problem. This leads to a robust method applicable to finite element problems discretized by unstructured meshes such as those from adaptive remeshing strategies. The method is based on specific properties of hierarchical quadratic bases. It can be combined with an algebraic multigrid (AMG) preconditioner or with other algebraic multilevel block factorizations. The algorithm can be accelerated by flexible Krylov subspace methods. We present some numerical results on the convection–diffusion and linear elasticity problems to illustrate the efficiency and the robustness of the presented algorithm. In these experiments, the performance of the proposed method is compared with that of an AMG preconditioner and other iterative solvers. Our approach requires less computing time and less memory storage. Copyright © 2010 John Wiley & Sons, Ltd.     

A constrained optimization approach to finite element mesh smoothing

The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the “smoothness” or “shape” of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a “smooth” final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method.     

Monotone iterative methods for the adaptive finite element solution of semiconductor equations

Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive finite element solution of semiconductor equations in terms of the Slotboom variables. The adaptive meshes are generated by the 1-irregular mesh refinement scheme. Based on these unstructured meshes and a corresponding modification of the Scharfetter–Gummel discretization scheme, it is shown that the resulting finite element stiffness matrix is an M-matrix which together with the Shockley–Read–Hall model for the generation–recombination rate leads to an existence–uniqueness–comparison theorem with simple upper and lower solutions as initial iterates. Numerical results of simulations on a MOSFET device model are given to illustrate the accuracy and efficiency of the adaptive and monotone properties of the present methods.     

Mesh and solver co‐adaptation in finite element methods for anisotropic problems

Mesh generation and algebraic solver are two important aspects of the finite element methodology. In this article, we are concerned with the joint adaptation of the anisotropic triangular mesh and the iterative algebraic solver. Using generic numerical examples pertaining to the accurate and efficient finite element solution of some anisotropic problems, we hereby demonstrate that the processes of geometric mesh adaptation and the algebraic solver construction should be adapted simultaneously. We also propose some techniques applicable to the co‐adaptation of both anisotropic meshes and linear solvers. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.     

Coupled FEM-BEM for elastoplastic contact problems using Lagrange multipliers

The coupling of the elastoplastic finite element and elastic boundary element methods for two-dimensional frictionless contact stress analysis is presented. Interface traction matching (boundary element approach), which involves the force terms in the finite element analysis being transformed to tractions, is chosen for the coupling method. The analysis at the contact region is performed by the finite element method, and the Lagrange multiplier approach is used to apply the contact constraints. Since the analyses of elastoplastic problems are non-linear and involve iterative solution, the reduced size of the final system of equations introduced by combining the two methods is very advantageous, especially for contact problems where the nature of the problem also involves an iterative scheme.     

Adaptive computation of smallest eigenvalues of self‐adjoint elliptic partial differential equations

We consider a new adaptive finite element (AFEM) algorithm for self‐adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finite‐dimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors, and use these for the adaptation process. We show that it is also possible to restrict to very few iterations of a Krylov subspace solver for the eigenvalue problem on coarse meshes. Several examples are presented to show that this new approach achieves much better complexity than the previous AFEM approaches which assume that the algebraic eigenvalue problem is solved to full accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Mesh-dependent stability for finite element approximations of parabolic equations with mass lumping

In this paper, based on some mesh-dependent estimates on the extreme eigenvalues of a general finite element system defined on a simplicial mesh, novel and sharp bounds on the permissible time step size are derived for the mass lumping finite element approximations of parabolic equations. The bounds are dependent not only on the mesh size but also on the mesh shape. These results provide guidance to the stability of numerical solutions of parabolic problems in relation to the unstructured geometric meshing. Numerical experiments on both uniform meshes and adaptive meshes are presented to validate the theoretical analysis.     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

An upwind-mixed method on changing meshes for two-phase miscible flow in porous media

Two-phase miscible flow in porous media is governed by a system of nonlinear partial differential equations. In this paper, the upwind-mixed method on dynamically changing meshes is presented for the problem in two dimensions. The pressure is approximated by a mixed finite element method and the concentration by a method which upwinds the convection and incorporates diffusion using an expanded mixed finite element method. The method developed is shown to obtain almost optimal rate error estimate. When the method is modified we can obtain the optimal rate error estimate that is well known for static meshes. The modification of the scheme is the construction of a linear approximation to the solution, which is used in projecting the solution from one mesh to another. Finally, numerical experiments are given.     

Singular solutions of the Poisson equation at edges of three‐dimensional domains and their treatment with a predictor–corrector finite element method

Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ? ?³ with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H¹(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions.     

A postprocessed flux conserving finite element solution

We propose a local postprocessing method to get a new finite element solution whose flux is conservative element‐wise. First, we use the so‐called polynomial preserving recovery (postprocessing) technique to obtain a higher order flux which is continuous across the element boundary. Then, we use special bubble functions, which have a nonzero flux only on one face‐edge or face‐triangle of each element, to correct the finite element solution element by element, guided by the above super‐convergent flux and the element mass. The new finite element solution preserves mass element‐wise and retains the quasioptimality in approximation. The method produces a conservative flux, of high‐order accuracy, satisfying the constitutive law. Numerical tests in 2D and 3D are presented.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1859–1883, 2017     

Adaptive finite element analysis of structures under transient dynamic loading using modal superposition

In this paper, the error estimation and adaptive strategy developed for the linear elastodynamic problem under transient dynamic loading based on the Z–Z criterion is utilized for 2D and plate bending problems. An automatic mesh generator based on “growth meshing” is utilized effectively for adaptive mesh refinement. Optimal meshes are obtained iteratively corresponding to the prescribed domain discretization error limit and for a chosen number of basis modes satisfying modal truncation errors. Numerous examples show the effectiveness of the integrated approach in achieving the target accuracy in finite element transient dynamic analysis.     

Development of a P2 element with optimal L2 convergence for biharmonic equation

It is well known that convergence rate of finite element approximation is suboptimal in the L² norm for solving biharmonic equations when P₂ or Q₂ element is used. The goal of this paper is to derive a weak Galerkin (WG) P₂ element with the L² optimal convergence rate by assuming the exact solution sufficiently smooth. In addition, our new WG finite element method can be applied to general mesh such as hybrid mesh, polygonal mesh or mesh with hanging node. The numerical experiments have been conducted on different meshes including hybrid meshes with mixed of pentagon and rectangle and mixed of hexagon and triangle.     

On using meshes of mixed element types in adaptive finite element analysis

Four different automatic mesh generators capable of generating either triangular meshes or hybrid meshes of mixed element types have been used in the mesh generation process. The performance of these mesh generators were tested by applying them to the adaptive finite element refinement procedure. It is found that by carefully controlling the quality and grading of the quadrilateral elements, an increase in efficiency over pure triangular meshes can be achieved. Furthermore, if linear elements are employed, an optimal hybrid mesh can be obtained most economically by a combined use of the mesh coring technique suggested by Lo and Lau and a selective removal of diagonals from the triangular element mesh. On the other hand, if quadratic elements are used, it is preferable to generate a pure triangular mesh first, and then obtain a hybrid mesh by merging of triangles.     

Static contact analysis of spiral bevel gear based on modified VFIFE (vector form intrinsic finite element) method

Since the intrinsic limitations of FEM (Finite element method) and lumped-mass method, we derive the formula of 8-node hexahedral element based on VFIFE (vector form intrinsic finite element method) method and applied it in contact analysis of gears. This paper proposed a new method to determine pure nodal deformation, which could simplify the computation compared to the traditional VFIFE method. Combining the VFIFE method and matching contact algorithm, we analyzed spiral bevel gear meshing problems. Spiral bevel models with two different mesh densities are calculated analyzed by the VFIFE method and FEM. Performance indicators of gears are extracted and compared, including contact forces, contact and bending stresses, contact stress patterns and loaded transmission errors. The results show that the VFIFE method has a stable performance and reliable accuracy under coarse or refined mesh conditions, while the FEM inaccurately calculates the contact stress of the coarse mesh model. The examples demonstrate that the proposed method could precisely analyze gear meshing problems with a coarse mesh model, which provides a new solution for gear mechanics.     

A priori error estimates for a coupled finite element method and mixed finite element method for a fluid-solid interaction problem

This paper presents a heterogeneous finite element method fora fluid–solid interaction problem. The method, which combinesa standard finite element discretization in the fluid regionand a mixed finite element discretization in the solid region,allows the use of different meshes in fluid and solid regions.Both semi-discrete and fully discrete approximations are formulatedand analysed. Optimal order a priori error estimates in theenergy norm are shown. The main difficulty in the analysis iscaused by the two interface conditions which describe the interactionbetween the fluid and the solid. This is overcome by explicitlybuilding one of the interface conditions into the finite elementspaces. Iterative substructuring algorithms are also proposedfor effectively solving the discrete finite element equations.     

求解两类Maxwell方程组棱元离散系统的快速算法和自适应方法

Maxwell方程组棱元离散系统的快速算法和自适应方法是当前计算电磁场中的研究热点和难点. 首先, 针对H(curl)椭圆方程组的棱元离散系统, 通过建立棱元空间的稳定性分解, 设计了相应的快速迭代法和高效预条件子, 并且证明了迭代算法的收敛率和预条件子的条件数均不依赖于模型参数和网格规模. 其次, 针对时谐Maxwell方程组的棱有限元方法, 利用离散的Helmholtz分解, 连续散度为零函数对离散散度为零函数的逼近性和对偶论证, 获得了在L²和H(curl)范数下的拟最优误差估计. 进而设计和分析了相应的两网格法. 最后, 分别针对变系数H(curl)椭圆方程组和不定时谐Maxwell方程组, 考虑了一种不需要标记振荡项和加密单元不需要满足“内节点” 性质的自适应棱有限元法(AEFEM), 并证明了AEFEM的收敛性. 进一步, 当初始网格和Dörfler标记策略参数满足一定的假设条件时, 利用AEFEM的收敛性、误差的整体下界和局部上界估计, 证明了AEFEM的拟最优复杂性.