n-sided polygonal hybrid finite elements with unified fundamental solution kernels for topology optimization

In topology optimization, the optimized design can be obtained based on spatial discretization of design domain using natural polygonal finite elements to reduce the influence of mesh geometry on topology optimization solutions. However, the natural polygonal finite elements require separate interpolants for each type of elements and involve troublesome domain integrals. In this study, an alternative n-sided polygonal hybrid finite element possessing multiple-node connection is formulated in a unified form to compress the checkerboard patterns caused by numerical instability in topology optimization. Different from the natural polygonal finite elements, the present polygonal hybrid finite elements involve two sets of independent displacement fields. The intra-element displacement field defined inside the element is approximated by the linear combination of the fundamental solution of the problem to achieve the purpose of the local satisfaction of the governing equations of the problem, but not the specific boundary conditions and the inter-element continuity conditions. To overcome such drawback, the inter-element displacement field defined over the entire element boundary is independently approximated by means of the conventional shape function interpolation. As a result, only line integrals along the element boundary are involved in the computation, whose dimension is reduced by one compared to the domain integrals in the natural polygonal finite elements, and more importantly, allowing us to flexibly construct any polygons from Voronoi tessellations in discretizing complex design domains using same fundamental solution kernels. Numerical results obtained indicate that the present n-sided polygonal hybrid finite elements can produce more accurate displacement solutions and smaller mean compliance, compared to the standard finite elements and the natural polygonal finite elements.     

A combined finite element and Marker and cell method for solving Navier-Stokes equations

Summary A new method is developed to approximate Navier-Stokes equations for incompressible fluids on polygonal domains. This method is based on a simple finite element approximation adapted to the Marker and Cell technique. Its main originality lies in the fact that the components of the velocity are approximated on distinct interlaced networks. The error analysis shows that this method is of order one.     

On the finite element approximation of a cascade flow problem

Summary The finite element analysis of a cascade flow problem with a given velocity circulation round profiles is presented. The nonlinear problem for the stream function with nonstandard boundary conditions is discretized by conforming linear triangular elements. We deal with the properties of the discrete problem and study the convergence of the method both for polygonal and nonpolygonal domains, including the effect of numerical integration.     

Finite element methods for semilinear elliptic problems with smooth interfaces

The purpose of this paper is to study the finite element method for second order semilinear elliptic interface problems in two dimensional convex polygonal domains. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with straight interface triangles [Numer. Math., 79 (1998), pp. 175–202]. For a finite element discretization based on a mesh which involve the approximation of the interface, optimal order error estimates in L ² and H ¹-norms are proved for linear elliptic interface problem under practical regularity assumptions of the true solution. Then an extension to the semilinear problem is also considered and optimal error estimate in H ¹ norm is achieved.     

Error Analysis of Local Refinements of Polygonal Domains

This paper gives a thorough analysis of the local refinement method on plane polygonal domains with special attention to the treatment of reentrant corner. Convergence rates of the finite element method under various norms are derived via a systematic treatment of the interpolation theory in weighted Sobolev spaces. It is proved that by refining the mesh suitably, the finite element approximations for problems with singularities achieve the same convergence rates as those for smooth solutions.     

The Dual Singular Function Method for 2D Boundary Integral Equations

The accuracy of standard boundary element methods for elliptic boundary value problems deteriorates if the boundary of the domain contains corners or if the boundary conditions change along the boundary. Here we first investigate the convergence behaviour of standard spline Galerkin approximation on quasi-uniform meshes for boundary integral equations on polygonal domains. It turns out, that the order of convergence depends on some constant describing the singular behaviour of solutions near corner points of the boundary. In order to recover the full order of convergence for the Galerkin approximation we propose the dual singular function method which is often used for improving the accuracy of finite element methods. The theoretical convergence results are confirmed and illustrated by a numerical example.     

角域上Green函数及其有限元解的一些估计

角域上Ｇｒｅｅｎ函数及其有限元解的一些估计黄云清（湘潭大学，湖南４１１１０５）林群（中国科学院系统科学研究所，北京１０００８０）１９８９年７月２２日收到，１９９０年１１月２７日收到修改稿。一、引言与主要结果设是平面角域，为所有角点集．为权函数．设，存...     

On the finite element approximation for maxwell’s problem in polynomial domains of the plane

The time-harmonic Maxwell boundary value problem in polygonal domains of R² is considered. The behaviour of the solution in the neighbourhood of nonregular boundary points is given and asymptotic error estimates in L₂- and in curl-div-norm for a finite element approximation of the solution are derived     

Residual‐based a posteriori error estimator for the mixed finite element approximation of the biharmonic equation

A reliable and efficient residual‐based a posteriori error estimator is derived for the Ciarlet‐Raviart mixed finite element method for the biharmonic equation on polygonal domains. The performance of the estimator is illustrated by numerical experiments. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Finite element methods and their convergence for elliptic and parabolic interface problems

In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in two-dimensional convex polygonal domains. Nearly the same optimal -norm and energy-norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical. Received July 7, 1996 / Revised version received March 3, 1997     

Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L₂ and H 1 . The convergence rate in the L_∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

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     

Shape calculus and finite element method in smooth domains

The use of finite elements in smooth domains leads naturally to polyhedral or piecewise polynomial approximations of the boundary. Hence the approximation error consists of two parts: the geometric part and the finite element part. We propose to exploit this decomposition in the error analysis by introducing an auxiliary problem defined in a polygonal domain approximating the original smooth domain. The finite element part of the error can be treated in the standard way. To estimate the geometric part of the error, we need quantitative estimates related to perturbation of the geometry. We derive such estimates using the techniques developed for shape sensitivity analysis.     

A preconditioner for the h-p version of the finite element method in two dimensions

Summary. A preconditioner, based on a two-level mesh and a two-level orthogonalization, is proposed for the - version of the finite element method for two dimensional elliptic problems in polygonal domains. Its implementation is in parallel on the subdomain level for the linear or bilinear (nodal) modes, and in parallel on the element level for the high order (side and internal) modes. The condition number of the preconditioned linear system is of order , where is the diameter of the -th subdomain, and are the diameter of elements and the maximum polynomial degree used in the subdomain. This result reduces to well-known results for the -version (i.e. ) and the -version (i.e. ) as the special cases of the - version. Received August 15, 1995 / Revised version received November 13, 1995     

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L² and H¹ norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.     

Partition of Unity for a Class of Nonlinear Parabolic Equation on Overlapping Non-Matching Grids

A class of nonlinear parabolic equation on a polygonal domain Ω  R2 is inves- tigated in this paper. We introduce a finite element method on overlapping non-matching grids for the nonlinear parabolic equation based on the partition of unity method. We give the construction and convergence analysis for the semi-discrete and the fully discrete finite element methods. Moreover, we prove that the error of the discrete variational problem has good approximation properties. Our results are valid for any spatial dimensions. A numerical example to illustrate the theoretical results is also given.     

Direct and inverse error estimates for finite elements with mesh refinements

The finite element method is used to solve a second order elliptic boundary value problem on a polygonal domain. Mesh refinements and weighted Besov spaces are used to obtain optimal error estimates and inverse theorems.Research performed while at the University of Maryland under a Fulbright fellowshipResearch supported in part by the Department of Energy under the contract E(40-1)3443Research supported in part by the National Institutes of Health under the grant 5R01-AM-20373     

Jacobi Polynomials,Weighted Sobolev Spaces and Approximation Results of Some Singularities

In this paper, we give some polynomial approximation results in a class of weighted Sobolev spaces, which are related to the Jacobi operator. We further give some embeddings of those weighted Sobolev spaces into usual ones and into spaces of continuous functions, in order to use the above approximation results in the p‐version (or the spectral method) of some finite or boundary element methods. Finally, two typical examples of the polynomial approximation of some singularities of boundary value problems in polygonal or polyhedral domains are presented.     

Discontinuous Galerkin finite element heterogeneous multiscale method for advection–diffusion problems with multiple scales

A discontinuous Galerkin finite element heterogeneous multiscale method is proposed for advection–diffusion problems with highly oscillatory coefficients. The method is based on a coupling of a discontinuous Galerkin discretization for an effective advection–diffusion problem on a macroscopic mesh, whose a priori unknown data are recovered from micro finite element calculations on sampling domains within each macro element. The computational work involved is independent of the high oscillations in the problem at the smallest scale. The stability of our method (depending on both macro and micro mesh sizes) is established for both diffusion dominated and advection dominated regimes without any assumptions about the type of heterogeneities in the data. Fully discrete a priori error bounds are derived for locally periodic data. Numerical experiments confirm the theoretical error estimates.     

Discontinuous Galerkin and mimetic finite difference methods for coupled Stokes–Darcy flows on polygonal and polyhedral grids

We study locally mass conservative approximations of coupled Darcy and Stokes flows on polygonal and polyhedral meshes. The discontinuous Galerkin (DG) finite element method is used in the Stokes region and the mimetic finite difference method is used in the Darcy region. DG finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces. Optimal convergence estimates for the numerical scheme are derived. Results from computational experiments supporting the theory are presented.