Asymptotic error estimates of a linearized projection-difference method for a differential equation with a monotone operator

We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem.     

Galerkin method for a third-order differential-operator equation

We study the Galerkin method for a third-order differential-operator equation with self-adjoint leading operator A and subordinate linear operator K(t) in a separable Hilbert space. We prove a theorem on the existence and uniqueness of a strong solution of the original problem. We derive estimates for the accuracy of the approximate solutions constructed by the Galerkin method. An application of the suggested method to the solution of a model problem is described.     

Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees p and q in space and time discretization, respectively. In the space discretization the nonsymmetric interior and boundary penalty approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”-and “ ”-norms, where ɛ ⩾ 0 is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization, we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution, the errors are of order O(h ^p + τ ^q ). The estimates hold true even in the hyperbolic case when ɛ = 0.     

An operator splitting method for monotone variational inequalities with a new perturbation strategy

In variational inequalities arising from applications such as engineering, economics and transportation, partial mappings are usually unknown, e.g., the demand function in traffic assignment problem. As a consequence, classical methods can not deal with this class of problems. On the other hand, the recently developed methods require restrictive conditions such as strong monotonicity of some mappings, which excludes many interesting applications. In this paper, we propose an operator splitting method with a new perturbation strategy for solving variational inequality problems with partially unknown mappings. Under the mild condition that the underlying mapping is monotone, we prove the global convergence of the method. We also report some preliminary numerical results which show that the new algorithm is also interesting from the numerical point of view.     

Wavelet Galerkin method for eigenvalue problem of a compact integral operator

We consider the approximation of eigenfunctions of a compact integral operator with a smooth kernel by the Galerkin method using wavelet bases. By truncating the Galerkin operator, we obtain a sparse representation of a matrix eigenvalue problem. We prove that the error bounds for the eigenvalues and for the distance between the spectral subspaces are of the orders O(nμ^-2nr) and O(μ^-nr), respectively, where μ⁻ⁿ denotes the norm of the partition and r denotes the order of the wavelet basis functions. By iterating the eigenvectors, we show that the error bounds for the eigenvectors are of the order O(nμ^-2nr). We illustrate our results with numerical results.     

Galerkin method for the linear-radiator integral equation

The article examines the general Galerkin method scheme for the singular integral equation in the problem of radiation from a finite-thickness linear radiator. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 150–158.     

A discrete Galerkin method for a hyperslngular boundary integral equation

Received on 14 August 1995. Revised on 20 August 1996. Consider solving the interior Neumann problem with D a simply-connected planar region and S=D a smooth curve.A double-layer potential is used to represent the solution,and it leads to the problem of solving a hypersingular integralequation. This integral equation is reformulated as a Cauchysingular integral equation. A discrete Galerkin method withtrigonometric polynomials is then given for its solution. Anerror analysis is given, and numerical examples complete thepaper.     

Piecewise-linear,discontinuous Galerkin method for a fractional diffusion equation

We use a piecewise-linear, discontinuous Galerkin method for the time discretization of a fractional diffusion equation involving a parameter in the range − 1 < α < 0. Our analysis shows that, for a time interval (0,T) and a spatial domain Ω, the error in L_￥((0,T);L₂(W))L_\infty\bigr((0,T);L_2(\Omega)\bigr) is of order k ^2 + α , where k denotes the maximum time step. Since derivatives of the solution may be singular at t = 0, our result requires the use of non-uniform time steps. In the limiting case α = 0 we recover the known O(k ²) convergence for the classical diffusion (heat) equation. We also consider a fully-discrete scheme that employs standard (continuous) piecewise-linear finite elements in space, and show that the additional error is of order h ²log(1/k). Numerical experiments indicate that our O(k ^2 + α ) error bound is pessimistic. In practice, we observe O(k ²) convergence even for α close to − 1.     

Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation

We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial discretization using continuous, piecewise-linear finite elements leads to an additional error term of order h ² max (1,logk ^− 1). Some simple numerical examples illustrate this convergence behaviour in practice. We thank the University of New South Wales for financial support provided by a Faculty Research Grant.     

A modified inexact operator splitting method for monotone variational inequalities

The Douglas–Peaceman–Rachford–Varga operator splitting methods (DPRV methods) are attractive methods for monotone variational inequalities. He et al. [Numer. Math. 94, 715–737 (2003)] proposed an inexact self-adaptive operator splitting method based on DPRV. This paper relaxes the inexactness restriction further. And numerical experiments indicate the improvement of this relaxation.      

Convergence of the Galerkin method for a wave equation with singular right-hand side

We consider analogs of the Galerkin method for a linear wave equation of the fifth order with generalized functions on the right-hand side. Theorems on the convergence of an approximate method, depending on the order of singularity of the right-hand side, are proved. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 778–786, June, 2006.     

A discontinuous Galerkin method for the Wigner-Fokker-Planck equation with a non-polynomial approximation space

A discontinuous Galerkin approach to the Wigner-Fokker-Planck equation, a model for quantum devices including environmental effects, is proposed. Evaluation of a pseudo-differential term is the main challenge. The approach can be applied to a variety of potential functions and uses general approximation spaces. Simulations using the method are in agreement with established analytic results and produce reasonable solutions for several potentials. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Differential inclusions with nonstationary maximal monotone operators

Institute for Control Problems. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 4, pp. 14–24, October–December, 1990.     

A discontinuous Galerkin method for the Cahn-Hilliard equation

The Cahn-Hilliard equation is modeled to describe the dynamics of phase separation in glass and polymer systems. A priori error estimates for the Cahn-Hilliard equation have been studied by the authors. In order to control accuracy of approximate solutions, a posteriori error estimation of the Cahn-Hilliard equation is obtained by discontinuous Galerkin method.     

A discontinuous Galerkin method for the Rosenau equation

A priori error estimates for the Rosenau equation, which is a K-dV like Rosenau equation modelled to describe the dynamics of dense discrete systems, have been studied by one of the authors. But since a priori error bounds contain the unknown solution and its derivatives, it is not effective to control error bounds with only a given step size. Thus we need to estimate a posteriori errors in order to control accuracy of approximate solutions using variable step sizes. A posteriori error estimates of the Rosenau equation are obtained by a discontinuous Galerkin method and the stability analysis is discussed for the dual problem. Numerical results on a posteriori error and wave propagation are given, which are obtained by using various spatial and temporal meshes controlled automatically by a posteriori error.     

Numerical analysis of a continuous Galerkin method for damped sine-Gordon equation

In this article, we discuss the numerical solution for the two-dimensional (2-D) damped sine-Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher-order accuracy in both space and time directions. The existence and uniqueness to the numerical solution are strictly proved and a priori error estimate in maximum-norm is given without any space–time grid conditions attached. Also, we prove that if the mesh in each time level is generated in a reasonable way, we can get the optimal order of convergence in both temporal and spatial variables. Finally, the convergence rates are presented and analyzed by some numerical experiments to illustrate the validity of the scheme.     

Partial inverse of a monotone operator

ForT a maximal monotone operator on a Hilbert spaceH andA a closed subspace ofH, the “partial inverse”T _A ofT with respect toA is introduced.T _A is maximal monotone. The proximal point algorithm, as it applies toT _A , results in a simple procedure, the “method of partial inverses”, for solving problems in which the object is to findx ∈ A andy ∈ A ^⊥ such thaty ∈ T(x). This method specializes to give new algorithms for solving numerous optimization and equilibrium problems.