This paper is concerned with the construction of high order mass-lumping finite elements on simplexes and a program for computing mass-lumping finite elements on triangles and tetrahedra. The polynomial spaces for mass-lumping finite elements, as proposed in the literature, are presented and discussed. In particular, the unisolvence problem of symmetric point-sets for the polynomial spaces used in mass-lumping elements is addressed, and an interesting property of the unisolvent symmetric point-sets is observed and discussed. Though its theoretical proof is still lacking, this property seems to be true in general, and it can greatly reduce the number of cases to consider in the computations of mass-lumping elements. A program for computing mass-lumping finite elements on triangles and tetrahedra, derived from the code for computing numerical quadrature rules presented in [7], is introduced. New mass-lumping finite elements on triangles found using this program with higher orders, namely 7, 8 and 9, than those available in the literature are reported.

A novel mesh deformation technique is developed based on the Delaunay graph mapping method and the inverse distance weighting (IDW) interpolation. The algorithm maintains the advantages of the efficiency of Delaunay graph mapping mesh deformation while it also possesses the ability of better controlling the near surface mesh quality. The Delaunay graph is used to divide the mesh domain into a number of sub-domains. On each sub-domain, the inverse distance weighting interpolation is applied, resulting in a similar efficiency as compared to the fast Delaunay graph mapping method. The paper will show how the near-wall mesh quality is controlled and improved by the new method

Motivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) and parameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for the L 2-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

The nonlinear Dirac equation is an important model in quantum physics with a set of conservation laws and a multi-symplectic formulation. In this paper, we propose energy-preserving and multi-symplectic wavelet algorithms for this model. Meanwhile, we evidently improve the efficiency of these algorithms in computations via splitting technique and explicit strategy. Numerical experiments are conducted during long-term simulations to show the excellent performances of the proposed algorithms and verify our theoretical analysis.

Through appropriate choices of elements in the underlying iterated function system, the methodology of fractal interpolation enables us to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as an α-fractal operator on , the space of all real-valued continuous functions defined on a compact interval I. With an eye towards connecting fractal functions with other branches of mathematics, in this paper we continue to investigate the fractal operator in more general spaces such as the space of all bounded functions and the Lebesgue space , and in some standard spaces of smooth functions such as the space of k-times continuously differentiable functions, Hölder spaces and Sobolev spaces . Using properties of the α-fractal operator, the existence of Schauder bases consisting of self-referential functions for these function spaces is established.

This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain. Two classes of artificial boundary conditions (ABCs) are designed, namely, nonlocal analog Dirichlet-to-Neumann-type ABCs (global in time) and high-order Padé approximate ABCs (local in time). These ABCs reformulate the original problem into an initial-boundary-value (IBV) problem on a bounded domain. For the global ABCs, we adopt a fast evolution to enhance computational efficiency and reduce memory storage. High order fully discrete schemes, both second-order in time and space, are given to discretize two reduced problems. Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.

Even though there are various fast methods and preconditioning techniques available for the simulation of Poisson problems, little work has been done for solving Poisson's equation by using the Helmholtz decomposition scheme. To bridge this issue, we propose a novel efficient algorithm to solve Poisson's equation in irregular two dimensional domains for electrostatics through a quasi-Helmholtz decomposition technique—the loop-tree basis decomposition. It can handle Dirichlet, Neumann or mixed boundary problems in which the filling media can be homogeneous or inhomogeneous. A novel point of this method is to first find the electric flux efficiently by applying the loop-tree basis functions. Subsequently, the potential is obtained by finding the inverse of the gradient operator. Furthermore, treatments for both Dirichlet and Neumann boundary conditions are addressed. Finally, the validation and efficiency are illustrated by several numerical examples. Through these simulations, it is observed that the computational complexity of our proposed method almost scales as , where N is the triangle patch number of meshes. Consequently, this new algorithm is a feasible fast Poisson solver.

The computation of integrals in higher dimensions and on general domains, when no explicit cubature rules are known, can be ”easily” addressed by means of the quasi-Monte Carlo method. The method, simple in its formulation, becomes computationally inefficient when the space dimension is growing and the integration domain is particularly complex. In this paper we present two new approaches to the quasi-Monte Carlo method for cubature based on nonnegative least squares and approximate Fekete points. The main idea is to use less points and especially good points for solving the system of the moments. Good points are here intended as points with good interpolation properties, due to the strict connection between interpolation and cubature. Numerical experiments show that, in average, just a tenth of the points should be used mantaining the same approximation order of the quasi-Monte Carlo method. The method has been satisfactory applied to 2 and 3-dimensional problems on quite complex domains.

In this paper, bymeans of a new recursive algorithm of non-tensor-product-typed divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimation is worked out, and an equivalence is obtained between high-order non-tensor-product-typed divided differences and high-order partial derivatives in the case of odd and even interpolating nodes, respectively. Thirdly, several numerical examples illustrate the recursive algorithms valid for the non-tensor-product-typed interpolating polynomials, and disclose that these polynomials change as the order of the interpolating nodes, although the node collection is invariant. Finally, from the aspect of computational complexity, the operation count with the bivariate polynomials presented is smaller than that with radial basis functions.

In this paper, the semilocal convergence for ameliorated super-Halley methods in Banach spaces is considered. Different from the results in [J. M. Gutiérrez and M. A. Hernández, Comput. Math. Appl. 36 (1998) 1–8], these ameliorated methods do not need to compute a second derivative, the computation for inversion is reduced and the $R$ -order is also heightened. Under a weaker condition, an existence–uniqueness theorem for the solution is proved.

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about O(h3), where h is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.

The present work constitutes a fraction of a more extensive study that is devoted to numerical methods in acoustics. More precisely, we address here the interpolation process, which is more and more frequently used in Computational Acoustics–whether it is for enabling multi-stage hybrid calculations, or for easing the proper handling of complex configurations via advanced techniques such as Chimera grids or Immersed Boundary Conditions. In that regard, we focus on high-order interpolation schemes, so as to analyze their intrinsic features and to assess their effective accuracy. Taking advantage of specific insights that had been previously achieved by the present authors regarding standard high-order interpolation schemes (of centered nature), we here focus on their so-called spectral-like optimized counterparts (of both centered and noncentered nature). The latter spectral-like optimized schemes are analyzed thoroughly thanks to dedicated theoretical developments, which allow highlighting better what their strengths and weaknesses are. Among others, the various ways such interpolation schemes can degrade acoustic signals they are applied to are carefully investigated from a theoretical point-of-view. Besides that, specific criteria that could help in optimizing interpolation schemes better are provided, along with generic rules about how to minimize the signal degradation induced by existing interpolation schemes, in practice.

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 C2-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 C1-quadratic splines over rectangular meshes are developed. Those bivariate splines are used to generate rational C1-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 C1–continuous connection schemes between two patches of the surfaces are presented.

A counterexample is constructed. It confirms that the error of conforming finite element solution is proportional to the coefficient jump, when solving interface elliptic equations. The Scott-Zhang operator is applied to a nonconforming finite element. It is shown that the nonconforming finite element provides the optimal order approximation in interpolation, in L2-projection, and in solving elliptic differential equation, independent of the coefficient jump in the elliptic differential equation. Numerical tests confirm the theoretical finding.

This paper extends an algorithm of P1-conservative interpolation on triangular meshes to tetrahedral meshes and thus constructs an approach of solution reconstruction for three-dimensional problems. The conservation property is achieved by local mesh intersection and the mass of a tetrahedron of the current mesh is calculated by the integral on its intersection with the background mesh. For each current tetrahedron, the overlapped background tetrahedrons are detected efficiently. A mesh intersection algorithm is proposed to construct the intersection of a current tetrahedron with the overlapped background tetrahedron and mesh the intersection region by tetrahedrons. A localization algorithm is employed to search the host units in background mesh for each vertex of the current mesh. In order to enforce the maximum principle and avoid the loss of monotonicity, correction of nodal interpolated solution on tetrahedral meshes is given. The performance of the present solution reconstruction method is verified by numerical experiments on several analytic functions and the solution of the flow around a sphere.

This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

This paper analyzes an abstract two-level algorithm for hybridizable discontinuous Galerkin (HDG) methods in a unified fashion. We use an extended version of the Xu-Zikatanov (X-Z) identity to derive a sharp estimate of the convergence rate of the algorithm, and show that the theoretical results also are applied to weak Galerkin (WG) methods. The main features of our analysis are twofold: one is that we only need the minimal regularity of the model problem; the other is that we do not require the triangulations to be quasi-uniform. Numerical experiments are provided to confirm the theoretical results.

We investigate the central moments of (regular) hexagons and derive accordingly a discrete approximation to definite integrals on hexagons. The seven-point cubature rule makes use of interior and neighbor center nodes, and is of fourth order by construction. The result is expected to be useful in two-dimensional (open-field) applications of integral equations or image processing.

Neighbour search (NS) is the core of any implementations of smoothed particle hydrodynamics (SPH). In this paper,we present an efficient neighbour search method based on the plane sweep (PW) algorithm with N being the number of SPH particles. The resulting method, dubbed the PWNS method, is totally independent of grids (i.e., purely meshfree) and capable of treating variable smoothing length, arbitrary particle distribution and heterogenous kernels. Several state-of-the-art data structures and algorithms, e.g., the segment tree and the Morton code, are optimized and implemented. By simply allowingmultiple lines to sweep the SPH particles simultaneously from different initial positions, a parallelization of the PWNS method with satisfactory speedup and load-balancing can be easily achieved. That is, the PWNS SPH solver has a great potential for large scale fluid dynamics simulations.

In two-phase flow simulations, a difficult issue is usually the treatment of surface tension effects. These cause a pressure jump that is proportional to the curvature of the interface separating the two fluids. Since the evaluation of the curvature incorporates second derivatives, it is prone to numerical instabilities. Within this work, the interface is described by a level-set method based on a discontinuous Galerkin discretization. In order to stabilize the evaluation of the curvature, a patch-recovery operation is employed. There are numerous ways in which this filtering operation can be applied in the whole process of curvature computation. Therefore, an extensive numerical study is performed to identify optimal settings for the patch-recovery operations with respect to computational cost and accuracy.