A refined mixed finite element method for the boussinesq equations in polygonal domains

This paper deals with the mixed formulation of the Boussinesqequations in two-dimensional polygonal domains and its numericalapproximation. The steady solution has a singular behaviournear the corner points so that we show that it belongs to appropriateweighted Sobolev spaces. Since uniform meshes lead to a slowconvergence rate, we derive appropriate refinement rules onthe meshes near the corner points in order to restore the quasi-optimalrate of convergence. A numerical test is finally presented whichconfirms the theoretical convergence rates. Received 26 November 1999. Accepted 25 August 2000.     

Local flux mimetic finite difference methods

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L ²-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.     

A Fast Numerical Method for Integral Equations of the First Kind with Logarithmic Kernel Using Mesh Grading

The aim of this paper is to develop a fast numerical method for two-dimensional boundary integral equations of the first kind with logarithm kernels when the boundary of the domain is smooth and closed. In this case, the use of the conventional boundary element methods gives linear systems with dense matrix. In this paper, we demonstrate that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. It will be demonstrated that this technique can increase the numerical efficiency significantly.     

Discrete boundary element methods on general meshes in 3D

Abstract. This paper is concerned with the stability and convergence of fully discrete Galerkin methods for boundary integral equations on bounded piecewise smooth surfaces in . Our theory covers equations with very general operators, provided the associated weak form is bounded and elliptic on , for some . In contrast to other studies on this topic, we do not assume our meshes to be quasiuniform, and therefore the analysis admits locally refined meshes. To achieve such generality, standard inverse estimates for the quasiuniform case are replaced by appropriate generalised estimates which hold even in the locally refined case. Since the approximation of singular integrals on or near the diagonal of the Galerkin matrix has been well-analysed previously, this paper deals only with errors in the integration of the nearly singular and smooth Galerkin integrals which comprise the dominant part of the matrix. Our results show how accurate the quadrature rules must be in order that the resulting discrete Galerkin method enjoys the same stability properties and convergence rates as the true Galerkin method. Although this study considers only continuous piecewise linear basis functions on triangles, our approach is not restricted in principle to this case. As an example, the theory is applied here to conventional “triangle-based” quadrature rules which are commonly used in practice. A subsequent paper [14] introduces a new and much more efficient “node-based” approach and analyses it using the results of the present paper. Received December 10, 1997 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

The focal geometry of circular and conical meshes

Circular meshes are quadrilateral meshes all of whose faces possess a circumcircle, whereas conical meshes are planar quadrilateral meshes where the faces which meet in a vertex are tangent to a right circular cone. Both are amenable to geometric modeling – recently surface approximation and subdivision-like refinement processes have been studied. In this paper we extend the original defining property of conical meshes, namely the existence of face/face offset meshes at constant distance, to circular meshes. We study the close relation between circular and conical meshes, their vertex/vertex and face/face offsets, as well as their discrete normals and focal meshes. In particular we show how to construct a two-parameter family of circular (resp., conical) meshes from a given conical (resp., circular) mesh. We further discuss meshes which have both properties and their relation to discrete surfaces of negative Gaussian curvature. The offset properties of special quadrilateral meshes and the three-dimensional support structures derived from them are highly relevant for computational architectural design of freeform structures. Another aspect important for design is that both circular and conical meshes provide a discretization of the principal curvature lines of a smooth surface, so the mesh polylines represent principal features of the surface described by the mesh.      

Infinitesimally flexible meshes and discrete minimal surfaces

We explore the geometry of isothermic meshes, conical meshes, and asymptotic meshes around the Christoffel dual construction of a discrete minimal surface. We present a discrete Legendre transform which realizes discrete minimal surfaces as conical meshes. Conical meshes turn out to be infinitesimally flexible if and only if their spherical image is isothermic, which implies that discrete minimal surfaces constructed in this way are infinitesimally flexible, and therefore possess reciprocal-parallel meshes. These are discrete minimal surfaces in their own right. In our study of relative kinematics of infinitesimally flexible meshes, we encounter characterizations of flexibility and isothermicity which are of incidence-geometric nature and are related to the classical Desargues configuration. The Lelieuvre formula for asymptotic meshes leads to another characterization of isothermic meshes in the sphere which is based on triangle areas.     

Spline Surfaces over Arbitrary Topological Meshes: Theoretical Analysis and Application

Based on polyhedral splines, some multivariate splines of different orders with given supports over arbitrary topological meshes are developed. Schemes for choosing suitable families of multivariate splines based on pre-given meshes are discussed. Those multivariate splines with inner knots and boundary knots from the related meshes are used to generate rational spline shapes with related control points. Steps for up to $C^2$-surfaces over the meshes are designed. The relationship among the meshes and their knots, the splines and control points is analyzed. To avoid any unexpected discontinuities and get higher smoothness, a heart-repairing technique to adjust inner knots in the multivariate splines is designed.With the theory above, bivariate $C^1$-quadratic splines over rectangular meshes are developed. Those bivariate splines are used to generate rational $C^1$-quadratic surfaces over the meshes with related control points and weights. The properties of the surfaces are analyzed. The boundary curves and the corner points and tangent planes, and smooth connecting conditions of different patches are presented. The $C^1$−continuous connection schemes between two patches of the surfaces are presented.     

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.     

A nine-point scheme with explicit weights for diffusion equations on distorted meshes

In this paper, for the structured quadrilateral mesh we derive a nine-point difference scheme which has five cell-centered unknowns and four vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear combination of the neighboring cell-centered unknowns, which reduces the scheme to a cell-centered one with a local stencil involving nine cell-centered unknowns. The coefficients in the linear combination are known as the weights and two types of new weights are proposed. These new weights are neither discontinuity dependent nor mesh topology dependent, have explicit expressions, can reduce to the one-dimensional harmonic-average weights on the nonuniform rectangular meshes, and moreover, are easily extended to the unstructured polygonal meshes and non-matching meshes. Both the derivation of the nine-point scheme and that of new weights satisfy the linearity preserving criterion. Numerical experiments show that, with these new weights, the nine-point difference scheme and its simple extension have a nearly second order accuracy on many highly distorted meshes, including structured quadrilateral meshes, unstructured polygonal meshes and non-matching meshes.     

Error analysis of staggered finite difference finite volume schemes on unstructured meshes

This work combines the consistency in lower‐order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured nonuniform meshes. This combined approach is first applied to a one‐dimensional elliptic boundary value problem on nonuniform meshes, and a first‐order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered Marker‐and‐Cell scheme for the two‐dimensional incompressible Stokes problem on unstructured meshes. A first‐order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1159–1182, 2017     

有限元网格的超收敛测度及其应用

给出线性有限元求解二阶椭圆问题的有限元网格超收敛测度及其应用.有限元超收敛经常是在具有一定结构的特殊网格条件下讨论的,而本文从一般网格出发,导出一种网格的范数用来描述超收敛所需要的网格条件以及超收敛的程度.并且通过对这种网格范数性质的考察,可以证明对于通常考虑的一些特殊网格的超收敛的存在性.更进一步,我们可以通过正则细分的方式在一般区域上也可以自动获得超收敛网格.最后给出相关的数值结果来验证本文的理论分析.     

Infinitesimally flexible meshes and discrete minimal surfaces

We explore the geometry of isothermic meshes, conical meshes, and asymptotic meshes around the Christoffel dual construction of a discrete minimal surface. We present a discrete Legendre transform which realizes discrete minimal surfaces as conical meshes. Conical meshes turn out to be infinitesimally flexible if and only if their spherical image is isothermic, which implies that discrete minimal surfaces constructed in this way are infinitesimally flexible, and therefore possess reciprocal-parallel meshes. These are discrete minimal surfaces in their own right. In our study of relative kinematics of infinitesimally flexible meshes, we encounter characterizations of flexibility and isothermicity which are of incidence-geometric nature and are related to the classical Desargues configuration. The Lelieuvre formula for asymptotic meshes leads to another characterization of isothermic meshes in the sphere which is based on triangle areas. Authors’ addresses: Johannes Wallner (corresponding author), Institut für Geometrie, TU Graz, Kopernikusgasse 24, A 8010 Graz, Austria; Helmut Pottmann, Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstr. 8-10/104, A 1040 Wien, Austria     

Progressive transmission of subdivision surfaces

Triangle meshes are a standard representation for surface geometry in computer graphics and virtual reality applications. To achieve high realism of the modeled objects, the meshes typically consist of a very large number of faces. For broadcasting virtual environments over low-bandwidth data connections like the Internet it is highly important to develop efficient algorithms which enable the progressive transmission of such large meshes. In this paper we introduce a special representation for storing and transmitting meshes with subdivision connectivity which allows random access to the detail information. We present algorithms for the decomposition and the reconstruction of subdivision surfaces. With this technique, the receiver can reconstruct smooth approximations of the original surface from a rather small amount of data received.     

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper, we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes. Different from the former scheme [J. Comput. Phys. 285(2015), 265-279] on uniform meshes, in this paper, in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme (UGKS), we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations. We can prove that the scheme is asymptotic preserving, and especially for the distorted quadrilateral meshes, a nine-point scheme [SIAM J. SCI. COMPUT. 30(2008), 1341-1361] for the diffusion limit equations is obtained, which is naturally reduced to standard five-point scheme for the orthogonal meshes. The numerical examples on distorted meshes are included to validate the current approach.     

Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods

For wave propagation in heterogeneous media, we compare numerical results produced by grid-characteristic methods on structured rectangular and unstructured triangular meshes and by a discontinuous Galerkin method on unstructured triangular meshes as applied to the linear system of elasticity equations in the context of direct seismic exploration with an anticlinal trap model. It is shown that the resulting synthetic seismograms are in reasonable quantitative agreement. The grid-characteristic method on structured meshes requires more nodes for approximating curved boundaries, but it has a higher computation speed, which makes it preferable for the given class of problems.     

On optimal triangular meshes for minimizing the gradient error

Summary Construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed. Using an appropriate linear coordinate transformation, rigorously optimal meshes for controlling the error in quadratic data functions are constructed. It is shown that the transformation can be generated as a curvilinear coordinate transformation for anyC data function with nonsingular Hessian matrix. Using this transformation, a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed. In particular, it is shown that equilibration of errors is not a sufficient condition for optimality. A comparison of meshes generated under several different criteria is made, and their equilibrating properties illustrated.This work was supported by the Natural Sciences and Engineering Research Council of Canada, by the Information Technology Research Centre, which is funded by the Province of Ontario, by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc., and through an appointment to the U.S. Department of Energy Postgraduate Research Program administered by Oak Ridge Associated Universities     

Quality mesh generation for molecular skin surfaces using restricted union of balls

Quality surface meshes for molecular models are desirable in the studies of protein shapes and functionalities. However, there is still no robust software that is capable to generate such meshes with good quality. In this paper, we present a Delaunay-based surface triangulation algorithm generating quality surface meshes for the molecular skin model. We expand the restricted union of balls along the surface and generate an ε-sampling of the skin surface incrementally. At the same time, a quality surface mesh is extracted from the Delaunay triangulation of the sample points. The algorithm supports robust and efficient implementation and guarantees the mesh quality and topology as well. Our results facilitate molecular visualization and have made a contribution towards generating quality volumetric tetrahedral meshes for the macromolecules.     

Nonconforming rotated Q_1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q1 element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

Three Dimension Quasi-Wilson Element for Flat Hexahedron Meshes

The well known Wilson‘s brick is only convergent for regular cuboid meshes. In this paper a quasi-Wilson element of three dimension is presented which is convergent for any hexahedron meshes. Meanwhile the element is anisotropic, that is it can be used to any flat hexahedron meshes for which the regular condition is unnecessary.     

CONVERGENCE ANALYSIS OF MORLEY ELEMENT ON ANISOTROPIC MESHES

The main aim of this paper is to study tile convergence of a nonconforming triangular plate element-Morley element under anisotropic meshes. By a novel approach, an explicit bound for the interpolation error is derived for arbitrary triangular meshes （which even need not satisfy the maximal angle condition and the coordinate system condition ）, the optimal consistency error is obtained for a family of anisotropically graded finite element meshes.