Stochastic Domain Decomposition for Time Dependent Adaptive Mesh Generation

The efficient generation of meshes is an important component in the numerical solution of problems in physics and engineering. Of interest are situations where global mesh quality and a tight coupling to the solution of the physical partial differential equation (PDE) is important. We consider parabolic PDE mesh generation and present a method for the construction of adaptive meshes in two spatial dimensions using stochastic domain decomposition that is suitable for an implementation in a multi- or many-core environment. Methods for mesh generation on periodic domains are also provided. The mesh generator is coupled to a time dependent physical PDE and the system is evolved using an alternating solution procedure. The method uses the stochastic representation of the exact solution of a parabolic linear mesh generator to find the location of an adaptive mesh along the (artificial) subdomain interfaces. The deterministic evaluation of the mesh over each subdomain can then be obtained completely independently using the probabilistically computed solutions as boundary conditions. A small scaling study is provided to demonstrate the parallel performance of this stochastic domain decomposition approach to mesh generation. We demonstrate the approach numerically and compare the mesh obtained with the corresponding single domain mesh using a representative mesh quality measure.     

数值解多维问题的外推与组合技术的若干新进展

本文综述近年来数值解多维问题的外推与组合技术的新进展，内容包括分裂外推及其在偏微分方程、多堆积分方程、多维数值积分中的应用；Ｃ．Ｚｅｎｇｅｒ的稀疏网格法与组合求解技术；以及解边界积分方程的组合方法，本文通过算例表明这些方法是非常有效的，是解多维问题的钥匙。     

On the Numerical Solution of Some Eikonal Equations: An Elliptic Solver Approach

The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute solutions, which are maximal and minimal in the variational sense. The approach in this paper relies on a variational argument involving penalty, a biharmonic regularization, and an operator-splitting-based time-discretization scheme for the solution of an associated initial-value problem. This approach allows the decoupling of the nonlinearities and differential operators. Numerical experiments are performed to validate this approach and investigate its convergence properties from a numerical viewpoint.     

A variational approach to optimal meshes

Summary. A variational approach for the optimization of triangular or tetrahedral meshes is presented. Starting from some very basic assumptions we will rigorously demonstrate that the functional controlling optimality is of a certain type related to energy functionals in non linear elasticity. It will be proved that these functionals attain their minima over admissible sets of mesh deformations which respect boundary conditions. In addition the injectivity of the deformed mesh is discussed. Thereby it is possible to construct suitable meshes for various numerical applications. Received March 14, 1994 / Revised version received August 8, 1994     

Numerical solution of the finite horizon stochastic linear quadratic control problem

The treatment of the stochastic linear quadratic optimal control problem with finite time horizon requires the solution of stochastic differential Riccati equations. We propose efficient numerical methods, which exploit the particular structure and can be applied for large‐scale systems. They are based on numerical methods for ordinary differential equations such as Rosenbrock methods, backward differentiation formulas, and splitting methods. The performance of our approach is tested in numerical experiments.     

Quadrature formulas descending from BS Hermite spline quasi-interpolation

Two new classes of quadrature formulas associated to the BS Boundary Value Methods are discussed. The first is of Lagrange type and is obtained by directly applying the BS methods to the integration problem formulated as a (special) Cauchy problem. The second descends from the related BS Hermite quasi-interpolation approach which produces a spline approximant from Hermite data assigned on meshes with general distributions. The second class formulas is also combined with suitable finite difference approximations of the necessary derivative values in order to define corresponding Lagrange type formulas with the same accuracy.     

Time-transformations for reversible variable stepsize integration

The development of a Sundman-type time-transformation for reversible variable stepsize integration of few-body problems is discussed. While a time-transformation based on minimum particle separation is suitable if the collisions only occur pairwise and isolated in time, the control of stepsize is typically much more difficult for a three-body close approach. Nonetheless, we find that a suitable choice of time-transformation based on particle separation can work quite well for certain types of three-body simulations, particularly those involving very steep repulsive walls. We confirm these observations using numerical examples from Lennard-Jones scattering. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analysis of a defect correction method for geometric integrators

We discuss a new variant of Iterated Defect Correction (IDeC), which increases the range of applicability of the method. Splitting methods are utilized in conjunction with special integration methods for Hamiltonian systems, or other initial value problems for ordinary differential equations with a particular structure, to solve the neighboring problems occurring in the course of the IDeC iteration. We demonstrate that this acceleration technique serves to rapidly increase the convergence order of the resulting numerical approximations, up to the theoretical limit given by the order of certain superconvergent collocation methods. This project was supported by the Special Research Program SFB F011 ‘AURORA’ of the Austrian Science Fund FWF.     

Two splitting schemes for nonstationary convection-diffusion problems on tetrahedral meshes

Two splitting schemes are proposed for the numerical solution of three-dimensional nonstationary convection-diffusion problems on unstructured meshes in the case of a full diffusion tensor. An advantage of the first scheme is that splitting is generated by the properties of the approximation spaces and does not reduce the order of accuracy. An advantage of the second scheme is that the resulting numerical solutions are nonnegative. A numerical study is conducted to compare the splitting schemes with classical methods, such as finite elements and mixed finite elements. The numerical results show that the splitting schemes are characterized by low dissipation, high-order accuracy, and versatility.     

高波数问题的超收敛性

 本文考虑求解Helmholtz方程的有限元方法的超逼近性质以及基于PPR后处理方法的超收敛性质.我们首先给出了矩形网格上的p-次元在收敛条件k（kh）^2p+1≤C₀下的有限元解和基于Lobatto点的有限元插值之间的超逼近以及重构的有限元梯度和精确解之间的超收敛分析.然后我们给出了四边形网格上的线性有限元方法的分析.这些估计都给出了与波数k和网格尺寸h的依赖关系.同时我们回顾了三角形网格上的线性有限元的超收敛结果.最后我们给出了数值实验并且结合Richardson外推进一步减少了误差.     

The use of geometric meshes in product integration Simpson''s rules

This paper concerns with the use of geometric meshes in product integration Simpson's rules for the numerical evaluation of weakly singular weighted integrals. We provide a criterion for selecting the mesh in order to achieve a desired accuracy with a low amount of computation. Numerical experiments indicate a reduction in computational costs with respect to the same rules with uniform meshes.     

Asset pricing for an affine jump‐diffusion model using an FD method of lines on nonuniform meshes

We present a novel numerical scheme for the valuation of options under a well‐known jump‐diffusion model. European option pricing for such a case satisfies a 1 + 2 partial integro‐differential equation (PIDE) including a double integral term, which is nonlocal. The proposed approach relies on nonuniform meshes with a focus on the discontinuous and degenerate areas of the model and applying quadratically convergent finite difference (FD) discretizations via the method of lines (MOL). A condition for observing the time stability of the fully discretized problem is given. Also, we report results of numerical experiments.     

Symmetric difference schemes of componentwise splitting and equivalent predictor-corrector scheme based on the Godunov method as applied to multidimensional gasdynamic simulation

An approach is described for improving the accuracy of numerical solutions to multidimensional gasdynamic problems produced by Godunov’s schemes. The basic idea behind the approach is to construct symmetric difference schemes based on splitting with respect to spatial variables with the subsequent transformation into equivalent predictor-corrector schemes. It is shown that the computation of “large” values by solving the one-dimensional Riemann problem at the interface of two neighboring cells leads to approximation errors in Godunov’s schemes. It is proposed to reconstruct large values so as to eliminate this source of errors. The time integration step in the modified schemes is consistent with that in the one-dimensional schemes and, on spatially uniform meshes, is 2 and 3 times larger than that in Godunov’s classical schemes for two- and three-dimensional problems, respectively. The numerical results obtained for test problems confirm the improvement of the accuracy of solutions produced by the modified schemes.     

Quadrature for meshless Nitsche's method

In this article, we study effect of numerical integration on Galerkin meshless method (GMM), applied to approximate solutions of elliptic partial differential equations with essential boundary conditions (EBC). It is well‐known that it is difficult to impose the EBC on the standard approximation space used in GMM. We have used the Nitsche's approach, which was introduced in context of finite element method, to impose the EBC. We refer to this approach as the meshless Nitsche's method (MNM). We require that the numerical integration rule satisfies (a) a “discrete Green's identity” on polynomial spaces, and (b) a “conforming condition” involving the additional integration terms introduced by the Nitsche's approach. Based on such numerical integration rules, we have obtained a convergence result for MNM with numerical integration, where the shape functions reproduce polynomials of degree k ≥ 1. Though we have presented the analysis for the nonsymmetric MNM, the analysis could be extended to the symmetric MNM similarly. Numerical results have been presented to illuminate the theoretical results and to demonstrate the efficiency of the algorithms.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 265–288, 2014     

A comparative study of ADI splitting methods for parabolic equations in two space dimensions

The main purpose of the paper is a numerical comparison of three integration methods for semi-discrete parabolic partial differential equations in two space variables. Linear as well as nonlinear,equations are considered. The integration methods are the well-known ADI method of Peaceman and Rachford, a global extrapolation scheme of the classical ADI method to order four and a fourth order, four-step ADI splitting method.     

A numerical solution of a one-dimensional blood flow model—moving grid approach

We consider a one-dimensional blood flow model suitable for larger arteries. It consists of a hyperbolic system of two coupled nonlinear equations. The model has already been successfully used in practice. Its numerical solution is usually achieved by means of an explicit Taylor–Galerkin scheme. We have proposed a different approach. The system can be transformed to characteristic directions emphasizing the physical nature of the problem. We solved this system by using an operator splitting on a moving grid.     

Stabilized DDFV schemes for stokes problem with variable viscosity on general 2D meshes

“Discrete Duality Finite Volume” schemes (DDFV for short) on general meshes are studied here for Stokes problems with variable viscosity with Dirichlet boundary conditions. The aim of this work is to analyze the well‐posedness of the scheme and its convergence properties. The DDFV method requires a staggered scheme, the discrete unknowns, the components of the velocity and the pressure, are located on different nodes. The scheme is stabilized using a finite volume analogue to Brezzi‐Pitkäranta techniques. This scheme is proved to be well‐posed on general meshes and to be first order convergent in a discrete H¹ ‐norm and a discrete L² ‐norm for respectively the velocity and the pressure. Finally, numerical experiments confirm the theoretical prediction, in particular on locally refined non conformal meshes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1666–1706, 2011     

A positive splitting method for mixed hyperbolic‐parabolic systems

In this article we present a method of lines approach to the numerical solution of a system of coupled hyperbolic—parabolic partial differential equations (PDEs). Special attention is paid to preserving the positivity of the solution of the PDEs when this solution is approximated numerically. This is achieved by using a flux‐limited spatial discretization for the hyperbolic equation. We use splitting techniques for the solution of the resulting large system of stiff ordinary differential equations. The performance of the approach applied to a biomathematical model is compared with the performance of standard methods. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 152–168, 2001     

MATHEMATICAL ANALYSIS FOR QUADRILATERAL ROTATED Q1 ELEMENT Ⅲ：THE EFFECT OF NUMERICAL INTEGRATION

This is the third part of the paper for the rotated Q1 nonconforming element on quadri-lateral meshes for general second order elliptic problems.Some optimal numerical formulas are presented and analyzed.The novelty is that it includes a formula with only two sam-pling points which excludes even a Q1 unisolvent set ,It is the optimal numerical integration formula over a quadrilateral mesh with least sampling points up to novw.     

Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations

The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.