Uniform Convergence Analysis for Singularly Perturbed Elliptic Problems with Parabolic Layers

In this paper,using Lin's integral identity technique,we prove the optimal uniform convergence θ(N_x~(-2)In~2N_x N_y~(-2)In~2N_y) in the L~2-norm for singularly per- turbed problems with parabolic layers.The error estimate is achieved by bilinear fi- nite elements on a Shishkin type mesh.Here N_x and N_y are the number of elements in the x- and y-directions,respectively.Numerical results are provided supporting our theoretical analysis.     

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS,Ⅱ

l)ThisworkwassupportedbyNWOthroughgrantIBo7-3Go12.BOUNDAarv^LUEPRoBLEMFORELLIPTICEQUMIONwiTHMIXEDBOUNDAavCONDITION1.IntroductionInthispedwesketchavarietyofspecialmethodswhichareusedforconstructinge-unifornilyconvergelltschemes-WeshaJldemonstrateamethodwhichachieveshaprovedaccuracyforsolvingsingularlyperturbedb0undaryvalueproblemforeiliPicequatiouswithparabolicboundarylayers-InSecti0n4weshallintroduceanaturalclass,B,oftritefferenceschemes,inwhich(bytheabovementi0nedaP…     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,III

1.IntroductionThesolution0fpartialdifferentiaJequationsthataresingularlyperturbedand/orhavediscontinu0usboundaryconditionsgenerallyhave0nlylimitedsmoothness.DuetothisfaCtdndcultiesaPpearwhenwesolvethesepr0blemsbynumericalmethods.Forexampleforregularparab0licequationswithdiscontinuousboundaryconditions,classicalmethods(FDMorFEM)onregularrectangulargridsd0n0tconvergeintheIoo-normonadomainthatincludesaneighbourhood0fthediscontinulty[8,9,4].Iftheparametermultiplyingthehighest-orderderivativeva…     

Discrete Approximations for Singularly Perturbed Boundary Value Problems with Parabolic Layers,I

In this series of three papers we study singularly perturbed (SP) boundary value problems for equations of elliptic and parabolic type. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids, we can construct difference schemes that allow approximation of the solution and the normalised diffusive flux uniformly with respect to the small parameter. We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges $ε$-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed. In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type; (iii) Problems for SP parabolic equation with discontinuous boundary conditions.     

Singularly Perturbed Parabolic Problems with Multidimensional Boundary Layers

The first boundary value problem for a multidimensional parabolic differential equation with a small parameter ε multiplying all derivatives is studied. A complete (i.e., of any order with respect to the parameter) regularized asymptotics of the solution is constructed, which contains a multidimensional boundary layer function that is bounded for x = (x¹, x₂) = 0 and tends to zero as ε → +0 for x ≠ 0. In addition, it contains corner boundary layer functions described by the product of a boundary layer function of the exponential type by a multidimensional parabolic boundary layer function.     

A Maximum Principle for Elliptic and Parabolic Equations with Oblique Derivative Boundary Problems

This paper prove a maximum principle for viscooity solutions of fully nonlinear, second order, uniformly elliptic and parabolic equations with oblique boundary value conditions.     

Efficient Collocation Schemes for Singular Boundary Value Problems

We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point.     

椭圆型方程组边值问题近似解的收敛性

本文研究了在矩形域内四阶系数不连续的椭圆型方程组边值问题近似解的收敛性问题,这对某一类弹性支承上的矩形板弯曲问题有参考价值。     

非线性椭圆型边值问题解的存在性

Hilbert空间方法被用于一类二阶半线性椭圆型边值问题并得出了某些条件下的解的存在性.     

General Linear Boundary Value Problems for Elliptic Operators with VMO Coefficients

We prove a priori estimates in Sobolev spaces for general linear elliptic boundary value problems with VMO coefficients.We give also results of existence and uniqueness of a solution.     

紧黎曼流形上的椭圆边界值问题

讨论一类非齐次非线性椭圆边界值问题.利用极大值原理证明了该问题解的梯度估计.作为它的应用得到了解的效率比估计.     

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Nonlinearities

Computational Mathematics and Mathematical Physics - A singularly perturbed parabolic equation $${{varepsilon }^{2}}left( {{{a}^{2}}frac{{{{partial }^{2}}u}}{{partial {{x}^{2}}}} -...     

Bernstein Polynomial Collocation Method for Elliptic Boundary Value Problems

We present a summary of recent developments in application of Bernstein polynomials to solution of elliptic boundary value problems with a pseudospectral method. Solution is approximated using Benstein polynomial interpolant defined at points of the extrema of Chebyshev polynomials i.e. the Chebyshev-Gauss-Lobatto (CGL) nodes. This approach brings impovement comparing to the Bernstein interpolation at equidistant nodes we used previously [1]. We show that this approach leads to spectral convergence and accuracy comparable to that of pseudospectral methods with orthogonal polynomials (Chebyshev, Legendre). The algorithm is implemented in open source library bernstein-poly , which is available online. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Convergence Results for Runge-Kutta Discretizations of Linear Nonautonomous Parabolic Problems

We analyze the convergence properties of implicit Runge-Kutta methods for the time-discretization of linear nonautonomous parabolic problems. We work in an abstract Banach space setting that covers standard parabolic initial-boundary value problems with time-dependent smooth coefficients. The Runge-Kutta methods are only assumed to be A()-stable, and variable stepsizes are allowed in the analysis. As the main result, we derive optimal convergence rates for Runge-Kutta discretizations of temporally smooth solutions.