Exponential splitting time integration for pseudospectral methods on moving meshes

Moving meshes are successfully used in many fields. Here we investigate how a recently proposed approach to combine the Strang splitting method for time integration with pseudospectral spatial discretization by orthogonal polynomials can be extended to include moving meshes. A double representation of a function (by coefficients of polynomial expansion and by values at the mesh nodes associated with a suitable quadrature formula) is an essential part of the numerical integration. Before numerical implementation the original PDE is transformed into a suitable form. The approach is illustrated on the linear heat transfer equation.     

油气资源数值模拟的变网格交替方向特征有限元格式和分析

石油科学在有机地球化学、石油的生成、运移、聚集研究取得重大进展,在评价油气资源时,对盆地发育史尤其对流体流动规律和受热变化历史的计算是非常重要的.其数学模型是三维空间非线性偶合偏微分方程组的初边值问题.从实际出发,考虑了流体的压缩性和三维问题大规模科学与工程计算的特征, 提出了一类变网格交替方向特征有限元格式,应用变分形式、算子分裂、广义$L^2$投影、能量方法、负模估计、微分方程先验估计的理论和技巧,得到最佳阶$L^2$误差估计. 此方法已成功应用到油气资源评估数值模拟的生产实践中,成功解决了这一重要问题.     

Shape regularity conditions for polygonal/polyhedral meshes,exemplified in a discontinuous Galerkin discretization

This article concerns shape regularity conditions on arbitrarily shaped polygonal/polyhedral meshes. In (J. Wang and X. Ye, A weak Galerkin mixed finite element method for second‐order elliptic problems, Math Comp 83 (2014), 2101–2126), a set of shape regularity conditions has been proposed, which allows one to prove important inequalities such as the trace inequality, the inverse inequality, and the approximation property of the L² projection on general polygonal/polyhedral meshes. In this article, we propose a simplified set of conditions which provides similar mesh properties. Our set of conditions has two advantages. First, it allows the existence of “small” edges/faces, as long as the shape of the polygon/polyhedron is regular. Second, coupled with an extra condition, we are now able to deal with nonquasiuniform meshes. As an example, we show that the discontinuous Galerkin method for Laplacian equations on arbitrarily shaped polygonal/polyhedral meshes, satisfying the proposed set of shape regularity conditions, achieves optimal rate of convergence. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 308–325, 2015     

Uniformly convergent difference schemes for a singularly perturbed third order boundary value problem

In this paper we consider a numerical approximation of a third order singularly perturbed boundary value problem by an upwind finite difference scheme on a Shishkin mesh. The behavior of the solution, and the stability of the continuous problem are discussed. The proof of the uniform convergence of the proposed numerical method is based on the strongly uniform stability and a weak consistency property of the discrete problem. Numerical experiments verify our theoretical results.     

Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes

A new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter‐free. Optimal order error estimates in a discrete H² norm is established for the corresponding WG finite element solutions. Error estimates in the usual L² norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003–1029, 2014     

Evaluating Local Approximations of the L2-Orthogonal Projection Between Non-Nested Finite Element Spaces

We present quantitative studies of transfer operators between finite element spaces associated with unrelated meshes. Several local approximations of the global $L^2$-orthogonal projection are reviewed and evaluated computationally. The numerical studies in 3D provide the first estimates of the quantitative differences between a range of transfer operators between non-nested finite element spaces. We consider the standard finite element interpolation, Clément's quasi-interpolation with different local polynomial degrees the global $L^2$-orthogonal projection, a local $L^2$-quasi-projection via a discrete inner product, and a pseudo-$L^2$-projection defined by a Petrov-Galerkin variational equation with a discontinuous test space. Understanding their qualitative and quantitative behaviors in this computational way is interesting per se; it could also be relevant in the context of discretization and solution techniques which make use of different non-nested meshes. It turns out that the pseudo-$L^2$-projection approximates the actual $L^2$-orthogonal projection best. The obtained results seem to be largely independent of the underlying computational domain; this is demonstrated by four examples (ball, cylinder, half torus and Stanford Bunny).     

抛物方程的一种广义差分法(有限体积法)

广义差分法自1982年被提出,至今已获得很大发展（见[1]或[10],这种方法在国际上被称为有限体积（元）法（见[8],[9]）,它的主要优点是保持物理量的局部守恒性.文[3],[5]分别将三角形网格上的椭圆型方程的广义差分法（有限体积法）（见[2],[4]）推广到抛物型方程.我们知道三角形网格与四边形网格是两种基本的分割空间区域的方法,实践上使用哪一种网格,要根据空间区域的几何形状而定.文[7],[6]讨论了一般四边形网上椭圆型方程的广义差分法.本文以抛物方程为模型,取试探函数空间为一般四边形剖分上的等参双线性元,检验函数空间为对偶剖分上的分片常数,导出了一种新的有效的广义差分算法（有限体积算法）,证明了半离散与全离散格式的最佳H1误差估计.遇到的主要困难是双线性形式a（uh,Πh＊uh）     

An Evaluation of HPF and MPI Approaches and Performance in Unstructured Finite Element Simulations

The High Performance Fortran (HPF) language and the Message Passing Interface (MPI) are two widely used methods to achieve parallelism on today's clusters and multiprocessor supercomputers. HPF is a distinct language providing extensions to Fortran 90/95 to express parallel execution paths and regions. MPI is a library of communication calls that can be inserted into modern high-level languages (C and Fortran). This paper discusses the use of the two approaches in a parallel finite element application for liquid composite manufacturing process modeling. The unstructured nature of the code provides an excellent opportunity to test both the computation and communication effectiveness of the two approaches. We discuss performance results based on implementations conducted on a modern massively parallel computing platform with a highly tuned processor interconnection network.     

基于自适应非结构化网格的VOF方法

本文提出了一种基于自适应非结构化网格的VOF算法,根据相界面网格的相函数值对相界面网格进行自适应细化与合并,通过基于非结构化网格的界面构造方法构造相界面,在适量增加网格单元数量的情况下提高了计算的精度。该方法随着时间及相界面的变化无需重新整体生成网格,算法效率较高。经典算例的验证结果表明,本文自适应网格方法计算得到的结...     

粘性不可压流体流动问题用直角坐标网格的贴体解法

研究一种新的全贴体的求解粘性不可压流体流动问题的非结构化直角坐标网格方法.该方法在于利用直角坐标网格但通过在边界附近保留不规则控制体,使得算法是完全贴体的.这有别于目前流行的各种非结构化直角坐标网格方法.通过对两个典型流动问题的计算对该数值方法进行验证.对比结果表明,本方法计算的结果与精确解和STAR-CD的结果在一定Re数和网格数时是很接近的,可以满足一定的精度要求,说明该数值计算方法是可行的.还对二维钝头体周围的流场进行了计算,计算的流场与STAR-CD的结果相当吻和,说明该算法还可计算较复杂的流动现象.