Zhaoliang Meng

Zhaoliang Meng, Zhongxuan Luo, Dongwoo Sheen

A new nonconforming rectangle element with cubic convergence for the energy norm is introduced. The degrees of freedom (DOFs) are defined by the twelve values at the three Gauss points on each of the four edges. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are eleven. The nonconforming element consists of $P_2\oplus \Span\{x^3y-xy^3\}$. We count the corresponding dimension for Dirichlet and Neumann boundary value problems of second-order elliptic problems. We also present the optimal error estimates in both broken energy and $L_2(Ø)$ norms. Finally, numerical examples match our theoretical results very well.

Zhaoliang Meng, Zhongxuan Luo, Dongwoo Sheen et al.

In this paper, we will give convergence analysis for a family of 14-node elements which was proposed by I. M. Smith and D. J. Kidger in 1992. The 14 DOFs are taken as the value at the eight vertices and six face-centroids. For second-order elliptic problem, we will show that among all the Smith-Kidger 14-node elements, Type 1, Type 2 and type 5 elements can get the optimal convergence order and Type 6 get lower convergence order. Motivated by the proof, we also present a new 14-node nonconforming element. If we change the DOFs into the value at the eight vertices and the integration value of six faces, we show that Type 1, Type 2 and Type 5 keep the optimal convergence order and Type 6 element improve one order accuracy which means that it also get optimal convergence order.

Zhaoliang Meng, Zhongxuan Luo

This paper will devote to construct a family of fifth degree cubature formulae for $n$-cube with symmetric measure and $n$-dimensional spherically symmetrical region. The formula for $n$-cube contains at most $n^2+5n+3$ points and for $n$-dimensional spherically symmetrical region contains only $n^2+3n+3$ points. Moreover, the numbers can be reduced to $n^2+3n+1$ and $n^2+n+1$ if $n=7$ respectively, the later of which is minimal.

Zhaoliang Meng, Zhongxuan Luo

In this paper we study the singularity of multivariate Hermite interpolation of type total degree. We present a method to judge the singularity of the interpolation scheme considered and by the method to be developed, we show that all Hermite interpolation of type total degree on $m=d+k$ points in $\R^d$ is singular if $d\geq 2k$. And then we solve the Hermite interpolation problem on $m\leq d+3$ nodes completely. Precisely, all Hermite interpolations of type total degree on $m\leq d+1$ points with $d\geq 2$ are singular; for $m=d+2$ and $m=d+3$, only three cases and one case can produce regular Hermite interpolation schemes, respectively. Besides, we also present a method to compute the interpolation space for Hermite interpolation of type total degree.

Xinchen Zhou, Zhaoliang Meng, Xin Fan et al.

This work is devoted to the high accuracy analysis of a discrete Stokes complex over rectangular meshes with a simple structure. The 0-form in the complex is a non $C^0$ nonconforming element space for biharmonic problems. This plate element contains only 12 degrees of freedom (DoFs) over a rectangular cell with a zeroth order weak continuity for the normal derivative, therefore only the lowest convergence order can be obtained by a standard consistency error analysis. Nevertheless, we prove that, if the rectangular mesh is uniform, an $O(h^2)$ convergence rate in discrete $H^2$-norm will be achieved. Moreover, based on a duality argument, it has an $O(h^3)$ convergence order in discrete $H^1$-norm if the solution region is convex. The 1-form and 2-form constitute a divergence-free pair for incompressible flow. We also show its higher accuracy than that derived from a usual error estimate under uniform partitions, which explains the phenomenon observed in our previous work. Numerical tests verify our theoretical results.

Xinchen Zhou, Zhaoliang Meng, Xin Fan et al.

This work introduces two 11-node triangular prism elements for 3D elliptic problems. The degrees of freedom (DoFs) of both elements are at the vertices and face centroids of a prism cell. The first element is $H^1$-nonconforming and works for second order problems, which achieves a second order convergence rate in discrete $H^1$-norm. The other is $H^2$-nonconforming and solves fourth order problems, with a first order convergence rate in discrete $H^2$-norm. Numerical examples verify our theoretical results.

Zhaoliang Meng, Zhongxuan Luo

Numerical integration formulas in $n$-dimensional Euclidean space of degree three are discussed. For the integrals with permutation symmetry we present a method to construct its third-degree integration formulas with $2n$ real points. We present a decomposition method and only need to deal with $n$ one-dimensional moment problems independently.

Xinchen Zhou, Zhaoliang Meng, Xin Fan et al.

This work proposes two nodal type nonconforming finite elements over convex quadrilaterals, which are parts of a finite element exact sequence. Both elements are of 12 degrees of freedom (DoFs) with polynomial shape function spaces selected. The first one is designed for fourth order elliptic singular perturbation problems, and the other works for Brinkman problems. Numerical examples are also provided.