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This paper deals with superconvergence phenomena in general grids when projectionbased approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the iterated Galerkin method, the Kantorovich method, and the iterated Kantorovich method. It is proved that the iterated Kantorovich approximation exhibits the best superconvergence rate if the right-hand side of the integral equation is nonsmooth. All error estimates are derived for an arbitrary grid without any uniformity or quasi-uniformity condition on it, and are formulated in terms of the data without any additional assumption on the solution. Numerical examples concern the equation governing transfer of photons in stellar atmospheres. The numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp, and confirm the superiority of the iterated Kantorovich scheme. [ABSTRACT FROM AUTHOR]

A recent study [D. N. Arnold, D. Boffi, and R. S. Falk, SIAM J. Numer. Anal., 42 (2005), pp. 2429-2451] reveals that convergence of finite element methods using H(div , O)-compatible finite element spaces deteriorates on nonaffine quadrilateral grids. This phenomena is particularly troublesome for the lowest-order Raviart-Thomas elements, because it implies loss of convergence in some norms for finite element solutions of mixed and least-squares methods. In this paper we propose reformulation of finite element methods, based on the natural mimetic divergence operator [M. Shashkov, Conservative Finite Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996], which restores the order of convergence. Reformulations of mixed Galerkin and leastsquares methods for the Darcy equation illustrate our approach. We prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator. Furthermore, we prove that standard and reformulated versions of the mixed Galerkin method lead to identical linear systems, but the two versions of the least-squares method are veritably different. The surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial, and that the lowest-order Raviart-Thomas elements are safe to use in this method. However, the breakdown in the least-squares method is real, and there one should use our proposed reformulation. [ABSTRACT FROM AUTHOR]

We develop a family of new interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model. This model is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. It has been recently shown that the convective part of this system is of a mixed hyperbolic-elliptic-type, which may cause severe instabilities when the studied system is solved by straightforward numerical methods. Therefore, the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original Keller-Segel model in the form of a convection-diffusionreaction system with a hyperbolic convective part. We then design interior penalty discontinuous Galerkin methods for the rewritten Keller-Segel system. Our methods employ the central-upwind numerical fluxes, originally developed in the context of finite-volume methods for hyperbolic systems of conservation laws. In this paper, we consider Cartesian grids and prove error estimates for the proposed high-order discontinuous Galerkin methods. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution. We also show that the blow-up time of the exact solution is bounded from above by the blow-up time of our numerical solution. In the numerical tests presented below, we demonstrate that the obtained numerical solutions have no negative values and are oscillation-free, even though no slope-limiting technique has been implemented. [ABSTRACT FROM AUTHOR]

The discontinuous Galerkin (DG) or local discontinuous Galerkin (LDG) method is a spatial discretization procedure for convection-diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters. The Lax-Wendroff time discretization procedure is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge-Kutta time discretizations. In this paper, we develop fluxes for the method of DG with Lax-Wendroff time discretization procedure (LWDG) based on different numerical fluxes for finite volume or finite difference schemes, including the first-order monotone fluxes such as the Lax-Friedrichs flux, Godunov flux, the Engquist-Osher flux etc. and the second-order TVD fluxes. We systematically investigate the performance of the LWDG methods based on these different numerical fluxes for convection terms with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one-dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we analyze the spatial and spectral superconvergence properties of one-dimensional hyperbolic conservation law by a discontinuous Galerkin (DG) method. The analyses combine classical mathematical arguments with MATLAB experiments. Some properties of the DG schemes are discovered using discrete Fourier analyses: superconvergence of the numerical wave numbers, Radau structure of the X spatial error. [Copyright &y& Elsevier]

Abstract: This paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty Galerkin methods. In particular, we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements. Numerical examples in all dimensions and for different polynomial degrees are presented. We show the numerical effects of loss of coercivity. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and analyze the continuous finite element method for a class of delay-differential equation with a variable term. At first for the case of the constant delay and based on an orthogonal expansion in an element we derive optimal superconvergence u − U =O(h p+2), p ⩾2 at the (p +1)-order Lobatto points and u − U =O(h 2p ) at the integer nodal points. Next we prove the stability of linear and quadratic finite element schemes for the case of constant delay. For the case of variable delay, by use of the approximative to the continuous delay in every subinterval we decompose the original problem into a series of subproblems with constant delay. Finally we give a recursive schemes which can be compute by every subproblem. [Copyright &y& Elsevier]

Abstract: In this paper, we provide a new meshless Petrov-Galerkin method, in which the trial space is generated by the global supported RBFs, the test space is generated by the compactly supported RBFs. The theoretical analysis and numerical results indicate that it is an efficient and accurate numerical method. [Copyright &y& Elsevier]

A finite element discretization is proposed and analyzed for a linear parabolic problems with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math., 79 (1998), pp. 175--202]. In this paper, we have used a finite element discretization, where interface triangles are assumed to be curved triangles instead of straight triangles as in classical finite element methods. Optimal order error estimates in L2 and H1 norms are shown to hold even if the regularity of the solution is low on the whole domain. While the continuous time Galerkin method is discussed for the spatially discrete scheme, the discontinuous Galerkin method is analyzed for the fully discrete scheme. The interfaces and boundaries of the domains are assumed to be smooth for our purpose. [ABSTRACT FROM AUTHOR]

In this paper we develop techniques for computing elementwise conservative approximations of the flux on element boundaries for the continuous Galerkin method. The technique is based on computing a correction of the average normal flux on an edge or face. The correction is a jump in a piecewise constant or linear function. We derive a basic algorithm which is based on solving a global system of equations and a parallel algorithm based on solving local problems on stars. The methods work on meshes with different element types and hanging nodes. We prove existence, uniqueness, and optimal order error estimates. Lastly, we illustrate our results by a few numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we present a meshless discretization technique for instationary convec- tion-diffusion problems. It is based on operator splitting, the method of characteristics, and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used as an h-version or a p-version. Even for general particle distributions, the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid. We discuss the implementational aspects of the proposed method. Furthermore, we present the results of numerical examples, where we considered instationary convection-diffusion, instationary diffusion, linear advection, and elliptic problems. [ABSTRACT FROM AUTHOR]