Pointwise error estimates for a stabilized Galerkin method: non-selfadjoint problems

Highly localized pointwise error estimates for a stabilized Galerkin method are provided for second-order non-selfadjoint elliptic partial differential equations. The estimates show a local dependence of the error on the derivative of the solution u and weak dependence on the global norm. The results in this paper are an extension of the previous pointwise error estimates for the self-adjoint problems. In order to provide pointwise error estimates in the presence of the first-order term in the differential equations, we prove that the stabilized Galerkin solution is higher order perturbation to the Ritz projection of the true solutions. Then, we proceed to obtain pointwise estimates using the so-called discrete Green’s function. Application to error expansion inequalities and a posteriori error estimators are briefly discussed.     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

A COMPLETE ANALYSIS FOR SOME A POSTERIORI ERROR ESTIMATES WITH THE FINITE ELEMENT METHOD OF LINES FOR A NONLINEAR PARABOLIC EQUATION

ABSTRACT

A posteriori error estimates for semidiscrete finite element methods for a nonlinear parabolic initial-boundary value problem are considered. The error estimates are obtained by solving local parabolic or elliptic equations for corrections to the solution on each element. The convergence results improve previous results where unnecessary assumptions are imposed on the approximate solution and the elliptic projection of the exact solution.     

A Multigrid Algorithm for Hybrid Joint PDF Simulations in Complex Geometries

In hybrid joint probability density function (joint PDF) algorithms for turbulent reactive flows the equations for the mean flow discretized with a classical grid based method (e.g. finite volume methods (FVM)) are solved together with a Monte Carlo (particle) method for the joint velocity composition PDF. When applied for complex geometries, the solution strategy for such methods which aims at obtaining a converged solution of the coupled problem on a sufficiently fine grid becomes very important. This paper describes one important aspect of this solution strategy, i.e. multigrid computing, which is well known to be very efficient for computing numerical solutions on fine grids. Two sets of grid based variables are involved: cell-centered variables from the FVM and node-centered variables, which denote the moments of the PDF extracted from the particle fields. Starting from a given multiblock grid environment first a new (refined or coarsened) grid is defined retaining the grid quality. The projection and prolongation operators are defined for the two sets of variables. In this new grid environment the particles are redistributed. The effectiveness of the multigrid algorithm is demonstrated. Compared to solely solving on the finest grid, convergence can be reached about one order of magnitude faster when using the multigrid algorithm in three stages. Computation time used for projection or prolongation is negligible. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

$L^1$-Stability and error estimates for approximate Hamilton-Jacobi solutions

Summary.   We study the -stability and error estimates of general approximate solutions for the Cauchy problem associated with multidimensional Hamilton-Jacobi (H-J) equations. For strictly convex Hamiltonians, we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors. We then demonstrate this general theory for two types of approximations: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type finite difference methods. If we let denote the `small scale' of such approximations (– the viscosity amplitude , the spatial grad-size , etc.), then our -error estimates are of , and are sharper than the classical -results of order one half, . The main building blocks of our theory are the notions of the semi-concave stability condition and -measure of the truncation error. The whole theory could be viewed as a multidimensional extension of the -stability theory for one-dimensional nonlinear conservation laws developed by Tadmor et. al. [34,24,25]. In addition, we construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a global projection operator. Here, we restrict our attention to linear projection operators (first-order schemes). We note, however, that our convergence theory applies equally well to nonlinear projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain -bounds on their associated truncation errors; -convergence of order one then follows. Second-order (central) Godunov-type schemes are also constructed. Numerical experiments are performed; errors and orders are calculated to confirm our -theory. Received April 20, 1998 / Revised version received November 8, 1999 / Published online August 24, 2000     

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology.The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical fluxes.By using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth enough.The theoretical results are proved for piecewise polynomials with any degree k 1.Finally,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy.Numerical results are provided to demonstrate these RKDG methods.     

SUPERAPPROXIMATION PROPERTIES OF THE INTERPOLATION OPERATOR OF PROJECTION TYPE AND APPLICATIONS

AbstractSome superapproximation and ultra-approximation properties in function, gradient and two-order derivative approximations are shown for the interpolation operator of projection type on two-dimensional domain. Then, we consider the Ritz projection and Ritz-Volterra projection on finite element spaces, and by means of the superapproximation elementary estimates and Green function methods, derive the superconvergence and ultraconvergence error estimates for both projections, which are also the finite element approximation solutions of the elliptic problems and the Sobolev equations, respectively.     

On the approximability and the selection of particle shape functions

Summary. Particle methods, also known as meshless or meshfree methods, have become popular in approximating solutions of partial differential equations, especially in the engineering community. These methods do not employ a mesh, or use it minimally, in the construction of shape functions. There is a wide variety of classes of shape functions that can be used in particle methods. In this paper, we primarily address the issue of selecting a class of shape functions, among this wide variety, that would yield efficient approximation of the unknown solution. We have also made several comments and observations on the order of convergence of the interpolation error, when these shape functions are used; specifically, we have shown that the interpolation error estimate, for certain classes of shape functions, may not indicate the actual order of convergence of the approximation error. Mathematics Subject Classification (2000):65N15, 65N30, 41A30This research was supported by NSF Grant # DMS-98-02367 and ONR Grant # N00014-99-1-0724     

Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems

We propose and study a posteriori error estimates for convection-diffusion-reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior-penalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction. We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using -conforming diffusive and convective flux reconstructions, thereby extending the previous work on pure diffusion problems. The resulting estimates are semi-robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local Péclet and Damköhler numbers. Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skew-symmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical experiments are presented to illustrate the theoretical results.     

Applications of Lobatto polynomials to an adaptive finite element method: A posteriori error estimates for HP-Adaptivity and Grid-to-Grid Interpolation

Summary.  Hp-adaptive finite element codes require methods for estimating the error at several spatial orders and for interpolating solutions between grids. Lobatto polynomial-based techniques are presented for both. An interpolation error-based error estimation strategy for a posteriori error estimates is generalized to yield asymptotically exact error estimates one order higher than the computed solution. The estimates involve high-order derivatives of the solution that must be approximated from the computed solution. Differentiating a ``Taylor-like' series for error in the Lobatto interpolant and using the weak form of the equations yields the correct derivative approximations. This leads to a more robust order selection strategy. Interpolation between grids is done over each element using the Lobatto interpolating polynomial. Explicit formulas for the inverse of the resulting Lobatto interpolation matrices are given. Computational results illustrate the theory. Received June 25, 2001 / Revised version received February 12, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65M15,65M20,65M60 This research was partially supported by NSF Grant #DMS-0196108.     

Globally optimal solutions of max–min systems

A variety of problems in computer science, operations research, control theory, etc., can be modeled as non-linear and non-differentiable max–min systems. This paper introduces the global optimization into such systems. The criteria for the existence and uniqueness of the globally optimal solutions are established using the high matrix, optimal max-only projection set and k ^s -control vector of max–min functions. It is also shown that the global optimization can be accomplished through the partial max-only projection representation with algebraic and combinatorial features. The methods are constructive and lead to an algorithm of finding all globally optimal solutions.     

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

Based on the pressure projection stabilized methods, the semi-discrete finite element approximation to the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions is considered in this paper. Because this class of boundary condition includes the subdifferential property, then the variational formulation is the Navier–Stokes type variational inequality problem. Using the regularization procedure, we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter \({\varepsilon}\), which means that the solution of the regularized problem converges to the solution of the Navier–Stokes type variational inequality problem as the parameter \({\varepsilon\longrightarrow 0}\). Moreover, some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data. The pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived.     

First Kind Cordial Volterra Integral Equations 1

For (pure) cordial Volterra integral equations of the first kind, we establish stability estimates of the solution in the scales of Banach spaces of C ^m -smooth functions on (0, T] and on [0, T]. This enables to estimate the error of polynomial or generalized polynomial approximate solutions which are easy to be determined.     

ELLIPTIC H^2-VOLTERRA PROJECTION AND THE H^1-GALERKIN METHODS FOR THE INTEGRO-DIFFERENTIAL EQUATIONS OF EVOLUTION

This paper studies H^1-Galerkin methods for the integro-differential equations of evolution. The elliptic H^2-Volterra projection is induced and then used in the derivations of error estimates for semi-discrete and full-discrete H^1-Galerkin methods.The optimal L^2, H^1 and H^2 norm error estimates are obtained.     

带弱奇异核的抛物型积分微分方程的非协调有限元方法

研究了带弱奇异核的抛物型积分微分方程的非协调有限元方法,在不需要Ritz-Volterra投影的情况下,在半离散和全离散的格式下分别得到了与协调有限元方法相同的误差估计.     

Compatible wavelet-Galerkin method to solve stokes interior problem with circular boundary

This article addresses Neumann boundary value interior problem of Stokes equations with circular boundary. By using natural boundary element method, the Stokes interior problem is reduced into an equivalent natural integral equation with a hyper-singular kernel, which is viewed as Hadamard finite part. Based on trigonometric wavelet functions, the compatible wavelet space is constructed so that it can serve as Galerkin trial function space. In proposed compatible wavelet-Galerkin method, the simple and accurate computational formulae of the entries in stiffness matrix are obtained by singularity removal technique. It is also proved that the stiffness matrix is almost a block diagonal matrix, and its diagonal sub-blocks all are both symmetric and circulant submatrices. These good properties indicate that a 2 ^J+3 × 2 ^J+3 stiffness matrix can be determined only by its 2 ^J + 3J + 1 entries. It greatly decreases the computational complexity. Some error estimates for the compatible wavelet-Galerkin projection solutions are established. Finally, several numerical examples are given to demonstrate the validity of the proposed approach.     

Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations

In this article, we propose two meshless collocation approaches for solving time dependent partial differential algebraic equations (PDAEs) in terms of the multiquadric quasi‐interpolation schemes. In presenting the process of the solution, the error is estimated. Furthermore, the comparisons on condition numbers of the collocation matrices using different methods and the sensitivity of the shape parameter c are given. With the use of the appropriate collocation points, the method for PDAEs with index‐2 is improved. The results show that the methods have some advantages over some known methods, such as the smaller condition numbers or more accurate solutions for PDAEs which has an modal index‐2 or an impulse solution with index‐2. Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 95–119, 2014     

Mortar element methods for parabolic problems

In this article a standard mortar finite element method and a mortar element method with Lagrange multiplier are used for spatial discretization of a class of parabolic initial‐boundary value problems. Optimal error estimates in L^∞(L²) and L^∞(H¹)‐norms for semidiscrete methods for both the cases are established. The key feature that we have adopted here is to introduce a modified elliptic projection. In the standard mortar element method, a completely discrete scheme using backward Euler scheme is discussed and optimal error estimates are derived. The results of numerical experiments support the theoretical results obtained in this article. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Finite element method for some nonlinear integro-differential equations

Abstract. This paper studies the finite element method for some nonlinear hyperbolic partial differential equations with memory and dampling terms. A Crank-Nicolson approximation for this kind of equations is presented. By using the elliptic Ritz Volterra projection,the analysis of the error estimates for the finite element numerical solutions and the optimal H1-norm error estimate are demonstrated.