Superconvergence of three dimensional Morley elements on cuboid meshes for biharmonic equations

In the present paper, superconvergence of second order, after an appropriate postprocessing, is achieved for three dimensional first order cuboid Morley elements of biharmonic equations. The analysis is dependent on superconvergence of second order for the consistency error and a corrected canonical interpolation operator, which help to establish supercloseness of second order for the corrected canonical interpolation. Then the final superconvergence is derived by a standard postprocessing. For first order nonconforming finite element methods of three dimensional fourth order elliptic problems, it is the first time that full superconvergence of second order is obtained without an extra boundary condition imposed on exact solutions. It is also the first time that superconvergence is established for nonconforming finite element methods of three dimensional fourth order elliptic problems. Numerical results are presented to demonstrate the validity of the theoretical results.     

Null sets and games in Banach spaces

The notion of Aronszajn-null sets generalizes the notion of Lebesgue measure zero in the Euclidean space to infinite dimensional Banach spaces. We present a game-theoretic approach to Aronszajn-null sets, establish its basic properties, and discuss some ensuing open problems.     

Null sets and games in Banach spaces

The notion of Aronszajn-null sets generalizes the notion of Lebesgue measure zero in the Euclidean space to infinite dimensional Banach spaces. We present a game-theoretic approach to Aronszajn-null sets, establish its basic properties, and discuss some ensuing open problems.     

On the Geometric Measure of Nodal Sets of Solutions

We present some general methods for the estimation of the local Hausdorff measure of nodal sets of solutions to elliptic and parabolic equations. Our main results (Theorems 3.1 and 4.1) improve previous results of Lin Fanghua in [1].     

On well‐posedness of nonclassical problems for elliptic equations

In the present paper, we consider nonclassical problems for multidimensional elliptic equations. A finite difference method for solving these nonlocal boundary value problems is presented. Stability, almost coercive stability and coercive stability for the solutions of first and second orders of approximation are obtained. The theoretical statements for the solutions of these difference schemes are supported by numerical examples for the two‐dimensional elliptic equations. Copyright © 2014 John Wiley & Sons, Ltd.     

Nodal Sets and Doubling Conditions in Elliptic Homogenization

This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients. We show that the(d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L_ε(u_ε) = 0 in a ball in Rdare bounded uniformly in ε 0. The proof relies on a uniform doubling condition and approximation of u_ε by solutions of the homogenized equation.     

The Dirichlet Problem for Semilinear Second-Order Degenerate Elliptic Equations and Applications to Stochastic Exit Time Control Problems

Abstract

We study nonlinear noncoercive elliptic problems with measure data, proving first that the global estimates already known when the problem is coercive are still true for noncoercive problems. We then prove new estimates, on sets far from the support of the singular part of the right-hand side, in the energy space associated to the operator, which entails additional regularity results on the solutions.     

A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation

Based on the homogeneous balance method,the Jacobi elliptic expansion method and the auxiliary equation method,the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations.New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple.The method is also valid for other (1+1)-dimensional and higher dimensional systems.     

Stability of the space identification problem for the elliptic‐telegraph differential equation

The present paper is devoted to study the space identification problem for the elliptic‐telegraph differential equation in Hilbert spaces with the self‐adjoint positive definite operator. The main theorem on the stability of the space identification problem for the elliptic‐telegraph differential equation is proved. In applications, theorems on the stability of three source identification problems for one dimensional with nonlocal conditions and multidimensional elliptic‐telegraph differential equations are established.     

On the differential variational inequalities of parabolic‐elliptic type

We consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality. The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness. Copyright © 2017 John Wiley & Sons, Ltd.     

Finite Element Approximation of Elliptic Dirichlet Optimal Control Problems

We develop a priori error analysis for the finite element Galerkin discretization of elliptic Dirichlet optimal control problems. The state equation is given by an elliptic partial differential equation and the finite dimensional control variable enters the Dirichlet boundary conditions. We prove the optimal order of convergence and present a numerical example confirming our results.     

Schauder estimates for elliptic operators with applications to nodal sets

In this paper we first give a priori estimates on asymptotic polynomials of solutions to elliptic equations at nodal points. This leads to a pointwise version of Schauder estimates. As an application we discuss the structure of nodal sets of solutions to elliptic equations with nonsmooth coefficients.     

Quantitative uniqueness for elliptic equations with singular lower order terms

We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second-order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.     

Stability of two‐dimensional flow in the two‐sided lid‐driven cavity

The one‐sided lid‐driven cavity has been studied quite extensively as benchmark problem in CFD. In the present study we discus the stability of two‐dimensional flows with respect to three‐dimensional perturbations of a more general lid‐driven cavity with two moving walls facing each other. We show that, in addition to the known elliptic‐instability branch around aspect ratio 1.5, another branch of the elliptic instability exists for aspect ratios smaller unity. For high and around unit aspect ratio the instabilities are found to be of centrifugal and quadripolar type, respectively. The structure of critical modes as well as the instability mechanism are addressed. Additionally, the numerical results are compared to experimental results of [5].     

Singularities of minima: a walk on the wild side of the Calculus of Variations

I will report on some recent developments concerning the problem of estimating the Hausdorff dimension of the singular sets of solutions to elliptic and variational problems. Emphasis will be given on some open issues. Connections with measure data problems will be outlined.     

Consistency,stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems

Summary. In an abstract framework we present a formalism which specifies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: in a two dimensional domain with Dirichlet boundary conditions. Received June 10, 1992 / Revised version received February 28, 1994     

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.     

Homogenized coefficients,quasiregular mappings and higher integrability of the gradients in two dimensional conductivity

The goal of this paper is to point out a connection between certain -closure problems relative to families of elliptic operators and the theory of planar quasiconformal mappings. In particular we consider a model (-closure) problem arising in two dimensional linear conductivity and we apply a recent result concerning the degree of integrability and the so-called measure dilatation of quasiconformal mappings to extract new information on the particular problem under consideration. Received June 13, 1994 / Accepted July 10, 1995     

Three solutions for a class of higher dimensional singular problems

In the present paper we establish the existence of three positive weak solutions for a quasilinear elliptic problem involving a singular term of the type \({u^{-\gamma}}\). As far as we know this is the first contribution in the higher dimensional case for arbitrary \({\gamma > 0}\).     

The Estimates Near the Boundary for Solutions on Monge-Ampere Equations

The present paper is concerned with the global C²-estimates of solutions to boundary value problems for degenerate elliptic. Monge-Ampere equations. Both of degeneracies of the boundary data and the right hand side of the equation are considered. In two dimensional case a result about smooth solutions is also contained.