Superconvergence of Solution Derivatives for the Shortley–Weller Difference Approximation of Poisson's Equation. II. Singularity Problems

Abstract

This is a continued analysis on superconvergence of solution derivatives for the Shortley–Weller approximation in Li (Li, Z. C., Yamamoto, T., Fang, Q. ([2003] Li, Z. C., Yamamoto, T. and Fang, Q. 2003. Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math., 152(2): 307–333.  [Google Scholar]): Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math. 152(2):307–333), which is to explore superconvergence for unbounded derivatives near the boundary. By using the stretching function proposed in Yamamoto (Yamamoto, T. ([2002] Yamamoto, T. 2002. Convergence of consistant and inconsistent finite difference schemes and an acceleration technique. J. Comp. Appl. Math., 140: 849–866. [Crossref], [Web of Science ®] , [Google Scholar]): Convergence of consistant and inconsistent finite difference schemes and an acceleration technique. J. Comp. Appl. Math. 140:849–866), the second order superconvergence for the solution derivatives can be established. Moreover, numerical experiments are provided to support the error analysis made. The analytical approaches in this article are non-trivial, intriguing, and different from Li, Z. C., Yamamoto, T., Fang, Q. ([2003] Li, Z. C., Yamamoto, T. and Fang, Q. 2003. Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math., 152(2): 307–333.  [Google Scholar]). This article also provides the superconvergence analysis for the bilinear finite element method and the finite difference method with nine nodes.     

On Periodic Solutions of Quasilinear Parabolic Equations with Boundary Conditions of Bitsadze–Samarskii Type

Doklady Mathematics - We consider a quasilinear parabolic boundary value problem with a nonlocal boundary condition of Bitsadze–Samarskii type. A theorem on the existence and uniqueness of a...     

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u _t  ? Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ?u/?n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = ?βu ^p , β ≥ 0, and g(x, u) = u ^q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T _b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T _b  ≤ T < τ. On the other hand, for the case f(x, u) = ?βu ^p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.     

Superconvergence of a Nonconforming Finite Element Approximation to Viscoelasticity Type Equations on Anisotropic Meshes

The main aim of this paper is to study the approximation to viscoelasticity type equations with a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes. The superclose property of the exact solution and the optimal error estimate of its derivative with respect to time are derived by using some novel techniques. Moreover, employing a postprocessing technique, the global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is studied.     

Semilinear Elliptic Equations with Uniform Blowup on the Boundary

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Fractional Differential Equations with Nonlocal Integral and Integer–Fractional-Order Neumann Type Boundary Conditions

We introduce a new concept of the coupling of nonlocal integral and integer–fractional-order Neumann type boundary conditions, and discuss the existence and uniqueness of solutions for a coupled system of fractional differential equations supplemented with these conditions. The existence of solutions is derived from Leray–Schauder’s alternative and Schauder’s fixed point theorem, while the uniqueness of solutions is established by means of Banach’s contraction mapping principle. The results obtained in this paper are well illustrated with the aid of examples.     

Stationary Solutions of Compressible Navier–Stokes Equations with Slip Boundary Conditions

We consider the Navier–Stokes equations for a compressible, viscous fluid with heat–conduction in a bounded domain of IR² or IR³. Under the assumption that the external force field and the external heat supply are small we prove the existence and local uniqueness of a stationary solution satisfying a slip boundary condition. For the temperature we assume a Dirichlet or an oblique boundary condition.     

On Type I Singularities of the Local Axi-Symmetric Solutions of the Navier–Stokes Equations

Local regularity of axially symmetric solutions to the Navier–Stokes equations is studied. It is shown that under certain natural assumptions there are no singularities of Type I.     

Landesman–Lazer Type Conditions and Multiplicity Results for Nonlinear Elliptic Problems with Neumann Boundary Values

We establish the existence and multiplicity of solutions for Steklov problems under non- resonance or resonance conditions using variational methods. In our main theorems, we consider a weighted eigenvalue problem of Steklov type.     

On Boundary Value Problems with the Shift for Nonlinear Elliptic Equations of Second Order

In this paper, we consider the boundary value problem with the shift for nonlinear uniformly elliptic equations of second order in a multiply connected domain,For this sake, we propose a modified boundary value problem for nonlinear elliptic systems of first order equations, and give a priori estimates of solutions for the modified boundary value problem. Afterwards we prove by using the Schauder fixedpoint theorem that this boundary value problem with some conditions has a solution. The result obtained is the generlization of the corresponding theorem on the Poincar boundary value problem.     

A Differential Boundary Operator of the Sturm–Liouville Type on a Semiaxis with Two-Point Integral Boundary Conditions

We prove that a differential boundary operator of the Sturm–Liouville type on a semiaxis with two-point integral boundary conditions that acts in the Hilbert space L ₂(0, ) is closed and densely defined. The adjoint operator is constructed. We also establish criteria for the maximal dissipativity and maximal accretivity of this operator.     

Solution Blowup for Nonlinear Equations of the Khokhlov–Zabolotskaya Type

We consider several nonlinear evolution equations sharing a nonlinearity of the form ?²u²/?t². Such a nonlinearity is present in the Khokhlov–Zabolotskaya equation, in other equations in the theory of nonlinear waves in a fluid, and also in equations in the theory of electromagnetic waves and ion–sound waves in a plasma. We consider sufficient conditions for a blowup regime to arise and find initial functions for which a solution understood in the classical sense is totally absent, even locally in time, i.e., we study the problem of an instantaneous blowup of classical solutions.     

On a Boundary Value Problem for a System of Differential–Difference Equations of Retarded Type

Differential Equations - We consider a control system described by a system of differential equations of retarded type with variable matrix coefficients and several delays. The relationship between...     

On the Uniqueness of the Solution of the Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph

For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.     

On the Convergence and Stability of a Pressure–Velocity Iteration of Chorin Type with Natural Boundary Conditions

Because of the implementation of numerical solution algorithms for the nonstationary Navier–Stokes equations of an incompressible fluid on massively parallel computers iterative methods are of special interest.

A red–black pressure–velocity iteration that allows an efficient parallelization based on a domain decomposition [3 G. Bärwolff and H. Schwandt ( 1996 ). A parallel domain decomposition algorithm in 3D turbulence modeling . In : Proceedings of the Int. Conf. on Parallel and Distributed Processing Techniques and Applications (PDPTA’96), August 8–11, 1996, Sunnyvale/CA . USA (Ed. H. Arabnia ), CSREA Press , pp. 44 – 45 . [Google Scholar]] will be analyzed in this paper.

We prove the equivalence of the pressure–velocity iteration (PUI) by Chorin/Hirt/Cook [1 A.J. Chorin ( 1968 ). Numerical Solution of the Navier–Stokes equations . Comp. Math. 22 : 745 – 762 .[Crossref] , [Google Scholar], 2 C.W. Hirt and J.L. Cook ( 1972 ). Calculating three-dimensional flows around structures and over rough terrain . J. Comp. Phy. 10 : 324 – 340 .[Crossref], [Web of Science ®] , [Google Scholar]] with a SOR iteration to solve a Poisson equation for the pressure. We show this on a 2D rectangle with some special outflow boundary conditions and Dirichlet data for the velocity elsewhere. This equivalence allows us to prove the convergence of that iteration scheme. We also discuss the stability of the occurring discrete Laplacian in discrete Sobolev spaces.     

Convergence Analysis of the Nonconforming Finite Element Discretization of Stokes and Navier–Stokes Equations with Nonlinear Slip Boundary Conditions

This work is concerned with the nonconforming finite approximations for the Stokes and Navier–Stokes equations driven by slip boundary condition of “friction” type. It is well documented that if the velocity is approximated by the Crouzeix–Raviart element of order one, whereas the discrete pressure is constant elementwise that the inequality of Korn does not hold. Hence, we propose a new formulation taking into account the curvature and the contribution of tangential velocity at the boundary. Using the maximal regularity of the weak solution, we derive a priori error estimates for the velocity and pressure by taking advantage of the enrichment mapping and the application of Babuska–Brezzi’s theory for mixed problems.