Interior estimates for the wavelet Galerkin method

Summary. In this paper we derive an interior estimate for the Galerkin method with wavelet-type basis. Such an estimate follows from interior Galerkin equations which are common to a class of methods used in the solution of elliptic boundary value problems. We show that the error in an interior domain can be estimated with the best order of accuracy possible, provided the solution is sufficiently regular in a slightly larger domain, and that an estimate of the same order exists for the error in a weaker norm (measuring the effects from outside the domain ). Examples of the application of such an estimate are given for different problems. Received May 17, 1995 / Revised version received April 26, 1996     

Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation error in the approximation of the nonlinear convective terms. The estimate of this error allows to analyse the error estimate of the method. The results obtained represent the completion and extension of the analysis from V. Dolej?í, M. Feistauer, Numer. Funct. Anal. Optim. 26 (2005), 349–383, where the truncation error in the approximation of the nonlinear convection terms was proved only in the case when the Dirichlet boundary condition on the whole boundary of the computational domain was considered.     

The fully Sinc‐Galerkin method for time‐dependent boundary conditions

The fully Sinc‐Galerkin method is developed for a family of complex‐valued partial differential equations with time‐dependent boundary conditions. The Sinc‐Galerkin discrete system is formulated and represented by a Kronecker product form of those equations. The numerical solution is efficiently calculated and the method exhibits an exponential convergence rate. Several examples, some with a real‐valued solution and some with a complex‐valued solution, are used to demonstrate the performance of this method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions

In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.     

Galerkin method for a nonlinear parabolic–elliptic system with nonlinear mixed boundary conditions

Materials which are heated by the passage of electricity are usually modeled by a nonlinear coupled system of two partial differential equations. The current equation is elliptic, while the temperature equation is parabolic. These equations are coupled one to another through the conductivities and the Joule effect. A computationally attractive discretization method is analyzed and shown to yield optimal error estimates in H¹. © 1993 John Wiley & Sons, Inc.     

On the effect of numerical integration in the Galerkin boundary element method

Summary. We discuss the effect of cubature errors when using the Galerkin method for approximating the solution of Fredholm integral equations in three dimensions. The accuracy of the cubature method has to be chosen such that the error resulting from this further discretization does not increase the asymptotic discretization error. We will show that the asymptotic accuracy is not influenced provided that polynomials of a certain degree are integrated exactly by the cubature method. This is done by applying the Bramble-Hilbert Lemma to the boundary element method. Received May 24, 1995     

Hypersingular kernel integration in 3D Galerkin boundary element method

We consider Cauchy singular and Hypersingular boundary integral equations associated with 3D potential problems defined on polygonal domains, whose solutions are approximated with a Galerkin boundary element method, related to a given triangulation of the boundary. In particular, for constant and linear shape functions, the most frequently used basis functions, we give explicit results of the analytical inner integrations and suggest suitable quadrature schemes to evaluate the outer integrals required to form the Galerkin matrix elements. These numerical indications are given after an analysis of the singularities arising in the whole integration process, which is valid also for shape functions of higher degrees.     

An augmented Galerkin procedure for the boundary integral method applied to mixed boundary value problems

Here we present the asymptotic error analysis for the boundary element approximation of the direct boundary integral equations for the plane mixed boundary value problem of the Laplacian. The boundary elements are defined by B-splines for the smooth parts of the boundary charges and additional singular functions at the collision points. The asymptotic error estimates include estimates for the stress intensity factors which occur as additional unknowns to be computed within the Galerkin scheme. The numerical analysis is based on the uniqueness of the problem, a coerciveness inequality, the triangular principal part and an extended shift theorem of the boundary integral operators.     

Spectral graph wavelet optimized finite difference method for solution of Burger's equation with different boundary conditions

In this paper, spectral graph wavelet optimized finite difference method (SPGWOFD) has been proposed for solving Burger's equation with distinct boundary conditions. Central finite difference approach is utilized for the approximations of the differential operators and the grid on which the numerical solution is obtained is chosen with the help of spectral graph wavelet. Four test problems (with Dirichlet, Periodic, Robin and Neumann's boundary conditions) are considered and the convergence of the technique is checked. For assessing the efficiency of the developed technique, the computational time taken by the developed technique is compared to that of the finite difference method. It has been observed that developed technique is extremely efficient.     

Multiplicative Schwarz algorithms for the p-version Galerkin boundary element method in 3D

We study a 2-level multiplicative Schwarz method for the p version Galerkin boundary element method for a weakly singular integral equation of the first kind in 3D. We prove that the rate of convergence of the multiplicative Schwarz operator for the p version grows only logarithmically in p and is independent of h.     

Additive Schwarz algorithms for the p version of the Galerkin boundary element method

Summary. We study some additive Schwarz algorithms for the version Galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first kind. Both non-overlapping and overlapping methods are considered. We prove that the condition numbers of the additive Schwarz operators grow at most as independently of h, where p is the degree of the polynomials used in the Galerkin boundary element schemes and h is the mesh size. Thus we show that additive Schwarz methods, which were originally designed for finite element discretisation of differential equations, are also efficient preconditioners for some boundary integral operators, which are non-local operators. Received June 15, 1997 / Revised version received July 7, 1998 / Published online February 17, 2000     

Optimizations of a fast multipole symmetric Galerkin boundary element method code

Numerical Algorithms - This paper presents some optimizations of a fast multipole symmetric Galerkin boundary element method code. Except general optimizations, the code is specially sped up for...     

On the energetic Galerkin boundary element method applied to interior wave propagation problems

We consider two-dimensional interior wave propagation problems with vanishing initial and mixed boundary conditions, reformulated as a system of two boundary integral equations with retarded potential. These latter are then set in a weak form, based on a natural energy identity satisfied by the solution of the differential problem, and discretized by the related energetic Galerkin boundary element method. Numerical results are presented and discussed.     

Error bounds for the Galerkin method applied to singular and nonsingular boundary value problems

Summary The purpose of this article is to obtainL ² and uniform norm error estimates for the Galerkin approximation of the solution of certain boundary value problems via a comparison with then-norm projection of the solution. In some cases, these estimates constitute an improvement over known results.This research was supported in part by AEC Grant (11-1)-2075.     

Left-definite boundary conditions of Sturm-Liouville problems when the interval shrinks to a point

On the condition that the interval of the problem shrinks to a point, we investigated the separated boundary conditions S α,β of left-definite Sturm-Liouville problem, and answered the following question: Is there a c 0 ∈ J such that S α,β is always left-definite or semi-left-definite for the Sturm-Liouville equation for each c ∈ (a, c 0 )?     

A convolution quadrature Galerkin boundary element method for the exterior Neumann problem of the wave equation

The numerical solution of the Neumann problem of the wave equation on unbounded three‐dimensional domains is calculated using the convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The mathematical analysis that has been built up for the Dirichlet problem is extended and developed for the Neumann problem, which is important for many modelling applications. Numerical examples are then presented for one of these applications, modelling transient acoustic radiation. Copyright © 2009 John Wiley & Sons, Ltd.     

On the implementation of boundary conditions for the method of lines

The method of lines for difference approximations of hyperbolic first order systems of partial differential equations is analyzed. The approximations are based on strictly semibounded difference operators including high order ones. The formulation of the ODE-system requires that the implementation of the boundary conditions is done carefully. We shall illustrate how different ways of implementation give rise to different stability properties. In particular, we derive a way of implementation that leads to an approximation that is strongly stable. It has been an open problem, whether for semidiscrete approximations with this strong stability property, the timestep for the ODE-solver is governed by the Cauchy problem. We present a counterexample showing that it is not. The analysis presented in this paper also serves as an illustration of the significant difference between different stability concepts.     

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd.