The discrete Galerkin method for integral equations

Iterated discrete polynomially based Galerkin methods

Solving a class of nonlinear boundary integral equations based on the meshless local discrete Galerkin (MLDG) method

Modelling and analysis framework for nonlinear dynamics of submerged tensioned cables

Four variants of the Galerkin method for integral equations of the form y = f+Ky, with K compact, are considered. They are the classical Galerkin method; the Kantorovich method (discussed recently by Schock), in which the inhomogeneous term f is treated exactly, and the Galerkin scheme applied to y−f; the iterated Galerkin method, obtained by substituting the Galerkin approximation into the right-hand side of the integral equation; and the similarly iterated Kantorovich method. The principal new result is that the iterated Kantorovich method converges faster than the Kantorovich method; and that the factor of improvement (the "superconvergence factor") has the same theoretical bound as that already known for iteration of the Galerkin method. Thus in the circumstances in which the Kantorovich modification is indicated (i.e. when y−f is smoother than y), the iterated Kantorovich method will generally be the most accurate of the four methods considered.

The Numerical Solution of Integral Equations on the Half-Line

We study a posteriori error estimates for the numerical approximations of state constrained optimal control problems governed by convection diffusion equations, regularized by Moreau---Yosida and Lavrentiev-based techniques. The upwind Symmetric Interior Penalty Galerkin (SIPG) method is used as a discontinuous Galerkin (DG) discretization method. We derive different residual-based error indicators for each regularization technique due to the regularity issues. An adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical examples are presented to illustrate the effectiveness of the adaptivity for both regularization techniques.

Efficient multistep procedures for time-stepping Galerkin methods for nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions are presented and analyzed. The procedures involve using a precoditioned iterative method for approximately solving the differet linear equations arising at each time step in a discrete time Galerkin method. Optimal order convergence rates are obtained for the iterative methods. Work estimates of almost optimal order are obtained.

Discrete multi-projection methods for eigen-problems of compact integral operators

We formulate and analyze new methods for the solution of a partial integrodifferential equation with a positive-type memory term. These methods combine the finite element Galerkin (FEG) method for the spatial discretization with alternating direction implicit (ADI) methods based on the Crank---Nicolson (CN) method and the second order backward differentiation formula for the time stepping. The ADI FEG methods are proved to be of optimal accuracy in time and in the $$L^2$$L2 norm in space. Furthermore, the analysis is extended to include an ADI CN FEG method with a graded mesh in time for problems with a nonsmooth kernel. Numerical results confirm the predicted convergence rates and also exhibit optimal spatial accuracy in the $$L^{\infty }$$L? norm.

We consider approximating the solution of the initial and boundary value problem for the Navier-Stokes equations in bounded two- and three-dimensional domains using a nonstandard Galerkin (finite element) method for the space discretization and the third order accurate, three-step backward differentiation method (coupled with extrapolation for the nonlinear terms) for the time stepping. The resulting scheme requires the solution of one linear system per time step plus the solution of five linear systems for the computation of the required initial conditions; all these linear systems have the same matrix. The resulting approximations of the velocity are shown to have optimal rate of convergence in L 2 {L^2} under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space.

Mathematics Department, University of Iowa, Iowa City, IA 52242, USA[Received 14 August 1995 and in revised form 20 August 1996]Consider solving the interior Neumann problemAu(P) = 0, PeD= /(/>), PeSwith D a simply-connected planar region and S = BD a smooth curve. Adouble-layer potential is used to represent the solution, and it leads to theproblem of solving a hypersingular integral equation. This integral equation isreformulated as a Cauchy singular integral equation. A discrete Galerkin methodwith trigonometric polynomials is then given for its solution. An error analysis isgiven, and numerical examples complete the paper.1. IntroductionLet D be a bounded open simply-connected region in the plane, and letits boundary S be sufficiently smooth. Consider the Neumann problem: findu 6 C

6. Bound on G In this section we adopt the approach of h [17] in order to bound IIGh 11 in (4.20). This approach is quite general, and the reader may wish to refer ahead to Section 7, which contains more specific examples of subspaces Sh. We distinguish between two cases: n (i) D is a region in iR (ii) D is a surface in in. In the first case, the modifications needed of Descloux's presentation are just notational; while in the second, they are minor in every respect. Thus we introduce the following notations and assumptions.

Numerial Methods for Volterra Integral Equations with Singular Kernels

Based on Landau-type transformation, a Stefan problem with nonlinear free boundary condition is transformed into a system consisting of parabolic equation and the ordinary differential equations. Semidiscrete approximations are constructed. Optimal orders of convergence of semidiscrete approximation inL2,H1 andH2 normed spaces are derived.

We consider a nonlinear parabolic system describing compressible miscible displacement in a porous medium[4]. Continuous time and discrete time Galerkin methods are introduced to approximate the solution, and optimalH 1-error estimates are obtained. This paper improves upon previously derived estimates in two aspects. Firstly, error estimates are given with no restrictions on the diffusion tensor. That is, we have included the effects of molecular diffusion and dispersion. Secondly, the complete compressible case is considered in the error analysis.