A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

In this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness of the algorithm.     

Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods

In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp-DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates.     

The polytopal structure of the tight-span of a totally split-decomposable metric

The tight-span of a finite metric space is a polytopal complex that has appeared in several areas of mathematics. In this paper we determine the polytopal structure of the tight-span of a totally split-decomposable (finite) metric. These metrics are a generalization of tree-metrics and have importance within phylogenetics. In previous work, we showed that the cells of the tight-span of such a metric are zonotopes that are polytope isomorphic to either hypercubes or rhombic dodecahedra. Here, we extend these results and show that the tight-span of a totally split-decomposable metric can be broken up into a canonical collection of polytopal complexes whose polytopal structures can be directly determined from the metric. This allows us to also completely determine the polytopal structure of the tight-span of a totally split-decomposable metric. We anticipate that our improved understanding of this structure may lead to improved techniques for phylogenetic inference.     

自适应间断Galerkin 有限元的多水平方法

本文讨论在自适应网格上间断Galerkin 有限元离散系统的局部多水平算法. 对于光滑系数和间断系数情形, 利用Schwarz 理论分析了算法的收敛性. 理论和数值试验均说明算法的收敛率与网格层数以及网格尺寸无关. 对强间断系数情形算法是拟最优的, 即收敛率仅与网格层数有关.     

A selective immersed discontinuous Galerkin method for elliptic interface problems

This article proposes a selective immersed discontinuous Galerkin method based on bilinear immersed finite elements (IFE) for solving second‐order elliptic interface problems. This method applies the discontinuous Galerkin formulation wherever selected, such as those elements around an interface or a singular source, but the regular Galerkin formulation everywhere else. A selective bilinear IFE space is constructed and applied to the selective immersed discontinuous Galerkin method based on either the symmetric or nonsymmetric interior penalty discontinuous Galerkin formulation. The new method can solve an interface problem by a rectangular mesh with local mesh refinement independent of the interface even if its geometry is nontrivial. Meanwhile, if desired, its computational cost can be maintained very close to that of the standard Galerkin IFE method. It is shown that the selective bilinear IFE space has the optimal approximation capability expected from piecewise bilinear polynomials. Numerical examples are provided to demonstrate features of this method, including the effectiveness of local mesh refinement around the interface and the sensitivity to the penalty parameters. Copyright © 2013 John Wiley & Sons, Ltd.     

Polytopality and Cartesian products of graphs

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.     

The classification of simplicial 3-spheres with nine vertices into polytopes and nonpolytopes

The classification of the 1296 (simplicial) 3-spheres with nine vertices into polytopal and nonpolytopal spheres, started earlier, is completed here. It is shown that there are 1142 polytopal and 154 nonpolytopal such spheres, and a fast procedure for their construction is described.     

A vertex-centered,dual discontinuous Galerkin method

This note introduces a new version of the discontinuous Galerkin method for discretizing first-order hyperbolic partial differential equations. The method uses piecewise polynomials that are continuous on a macroelement surrounding the nodes in the unstructured mesh but discontinuous between the macroelements. At lowest order, the method reduces to a vertex-centered finite-volume method with control volumes based on a dual mesh, and the method can be implemented using an edge-based data structure. The method provides therefore a strategy to extend existing vertex-centered finite-volume codes to higher order using the discontinuous Galerkin method. Preliminary tests on a model linear hyperbolic equation in two-dimensional indicate a favorable qualitative behavior for nonsmooth solutions and optimal convergence rates for smooth solutions.     

A hybrid high-order formulation for a Neumann problem on polytopal meshes

In this work, we study a hybrid high-order (HHO) method for an elliptic diffusion problem with Neumann boundary condition. The proposed method has several features, such as: (a) the support of arbitrary approximation order polynomial at mesh elements and faces on polytopal meshes, (b) the design of a local (element-wise) potential reconstruction operator and a local stabilization term, that weakly enforces the matching between local element- and face-based on degrees of freedom, and (c) cheap computational cost, thanks to static condensation and compact stencil. We prove the well-posedness of our HHO formulation, and obtain the optimal error estimates, according to previous study. Implementation aspects are thoroughly discussed. Finally, some numerical examples are provided, which are in agreement with our theoretical results.     

Polytopal complexes: maps,chain complexes and… necklaces

The notion of polytopal map between two polytopal complexes is defined. This definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the associated chain complexes. One uses this new tool to give the first combinatorial proof of the splitting necklace theorem of Alon.     

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.     

Polytopal estimate of Mirkovi?–Vilonen polytopes lying in a Demazure crystal

In this paper, we give a polytopal estimate of Mirkovi?–Vilonen polytopes lying in a Demazure crystal in terms of Minkowski sums of extremal Mirkovi?–Vilonen polytopes. As an immediate consequence of this result, we provide a necessary (but not sufficient) polytopal condition for a Mirkovi?–Vilonen polytope to lie in a Demazure crystal.     

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.     

Polytopal linear retractions

We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general.

This paper is a part of the project started by the authors in 1999, where we investigate the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable `polytopal' generalization of linear algebra (and, subsequently, that of algebraic -theory).

     

晶体微观结构的数值计算

晶体微观结构是晶体材料在特定物理条件下其多个能量极小平衔态在空间形成的某种微尺度的规则分布．几何非线性的连续介质力学理论可以用能量极小化原理来解释晶体微观结构的形成，并用Young测度来刻画平衡态各变体在空间的概率分布．定性的理解与定量地分析和计算晶体材料的微观结构对于发展和改进高级晶体功能材料，如形状记忆合金、铁电体、磁至伸缩材料等，有重要的意义．本文回顾了近年来晶体微观结构数值计算方面的最新进展．介绍了计算晶体微观结构的几种数值方法及有关的数值分析结果。     

The polytopal association scheme

Summary The polytopal association scheme for PBIB designs is introduced and studied utilizing the concept of clustering of treatments.     

On the Universal Partition Theorem for 4-Polytopes

By means of sign-patterns any finite family of polynomials induces a decomposition of R ⁿ into basic semialgebraic sets. In case of integer coefficients the latter decomposition roughly appears to be a partition into realization spaces of 4 -polytopes. The latter is stated by the Universal Partition Theorem for 4 -polytopes by Richter-Gebert. The present paper presents a different proof. As its main tool, the von Staudt polytope is introduced. The von Staudt polytope constitutes the polytopal equivalent of the well-known von Staudt constructions for point configurations. With the aid of the von Staudt polytope the original ideas of universality theory can be directly applied to the polytopal case. Moreover, a new method for representing real values (on a computation line) by polytopal means is presented. This method implies a bundling strategy in order to duplicate the encoded information. Based on this approach, the following complexity result is obtained. The incidence code of a polytope, exhibiting a realization space equivalent to a given semialgebraic set, can be computed in the same time that it requires to generate the defining polynomial system. Received December 19, 1995, and in revised form December 16, 1996, April 28, 1997, and September 10, 1997.     

Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation

In the construction of nine point scheme,both vertex unknowns and cell-centered unknowns are introduced,and the vertex unknowns are usually eliminated by using the interpolation of neighboring cell-centered unknowns,which often leads to lose accuracy.Instead of using interpolation,here we propose a different method of calculating the vertex unknowns of nine point scheme,which are solved independently on a new generated mesh.This new mesh is a Vorono¨i mesh based on the vertexes of primary mesh and some additional points on the interface.The advantage of this method is that it is particularly suitable for solving diffusion problems with discontinuous coeffcients on highly distorted meshes,and it leads to a symmetric positive definite matrix.We prove that the method has first-order convergence on distorted meshes.Numerical experiments show that the method obtains nearly second-order accuracy on distorted meshes.     

On the coupling of finite volume and discontinuous Galerkin method for elliptic problems

The coupling of cell-centered finite volume method with primal discontinuous Galerkin method is introduced in this paper for elliptic problems. Convergence of the method with respect to the mesh size is proved. Numerical examples confirm the theoretical rates of convergence. Advantages of the coupled scheme are shown for problems with discontinuous coefficients or anisotropic diffusion matrix.     

Fitted mesh method for singularly perturbed reaction-convection-diffusion problems with boundary and interior layers

A robust numerical method for a singularly perturbed secondorder ordinary differential equation having two parameters with a discontinuous source term is presented in this article. Theoretical bounds are derived for the derivatives of the solution and its smooth and singular components. An appropriate piecewise uniform mesh is constructed, and classical upwind finite difference schemes are used on this mesh to obtain the discrete system of equations. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are provided to illustrate the convergence of the numerical approximations.