A comparison of multilevel methods for total variation regularization.
A comparison of some a posteriori error estimates for fourth order problems
A lot of papers and books analyze analytical a posteriori error estimates from the point of view of robustness, guaranteed upper bounds, global efficiency, etc. At the same time, adaptive finite element methods have acquired the principal position among algorithms for solving differential problems in many physical and technical applications. In this survey contribution, we present and compare, from the viewpoint of adaptive computation, several recently published error estimation procedures for...
A comparison of some efficient numerical methods for a nonlinear elliptic problem
The aim of this paper is to compare and realize three efficient iterative methods, which have mesh independent convergence, and to propose some improvements for them. We look for the numerical solution of a nonlinear model problem using FEM discretization with gradient and Newton type methods. Three numerical methods have been carried out, namely, the gradient, Newton and quasi-Newton methods. We have solved the model problem with these methods, we have investigated the differences between them...
A comparison of the accuracy of the finite-difference solution to boundary value problems for the Helmholtz equation obtained by direct and iterative methods
The development of iterative methods for solving linear algebraic equations has brought the question of when the employment of these methods is more advantageous than the use of the direct ones. In the paper, a comparison of the direct and iterative methods is attempted. The methods are applied to solving a certain class of boundary-value problems for elliptic partial differential equations which are used for the numerical modeling of electromagnetic fields in geophysics. The numerical experiments...
A complement to the Fredholm theory of elliptic systems on bounded domains.
A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials
The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank-r convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank-r convex forms arises. In the present paper, we define the concept of extremal 2-forms and characterize them in the rotationally invariant jointly...
A condition on the potential for the existence of doubly periodic solutions of a semi-linear fourth-order partial differential equation.
A Conjugate Gradient Method and a Multigrid Algorithm for Morley' s Finite Element Approximation of the Biharmonic Equation.
A conservative particle approximation for a boundary advection-diffusion problem
A construction of a parametrix for an elliptic boundary value problem.
A construction of parametrices of elliptic boundary value problems
A continuity result for a quasilinear elliptic free boundary problem
We investigate a two dimensional quasilinear free boundary problem, and show that the free boundary is a union of graphs of continuous functions.
A convergence result and numerical study for a nonlinear piezoelectric material in a frictional contact process with a conductive foundation
We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky's law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the...
A convergent adaptive finite element method with optimal complexity.
A Counterexample in Elliptic Regularity Theory.
A counterexample to an endpoint bilinear Strichartz inequality.
A counterexample to Schauder estimates for elliptic operators with unbounded coefficients
We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space of . We show that for a particular initial datum, which is Lipschitz continuous and bounded on , the second derivative of the classical solution is not uniformly continuous on . In particular this implies that the well known maximal Hölder-regularity results fail in general for Dirichlet problems in unbounded domains involving unbounded coefficients.
A Counter-Example to the Boundary Regularity of Solutions to Elliptic Quasilinear Systems.
A counterexample to the “hot spots" conjecture.
A counterexample to the -Hodge decomposition
We construct a bounded domain with the cone property and a harmonic function on Ω which belongs to for all 1 ≤ p < 4/3. As a corollary we deduce that there is no -Hodge decomposition in for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in for all p > 4.