A posteriori estimators for nonlinear elliptic partial differential equations

Many works have reported results concerning the mathematical analysis of the performance of a posteriori error estimators for the approximation error of finite element discrete solutions to linear elliptic partial differential equations. For each estimator there is a set of restrictions defined in such a way that the analysis of its performance is made possible. Usually, the available estimators may be classified into two types, i.e., the implicit estimators (based on the solution of a local problem) and the explicit estimators (based on some suitable norm of the residual in a dual space). Regarding the performance, an estimator is called asymptotically exact if it is a higher-order perturbation of a norm of the exact error. Nowadays, one may say that there is a larger understanding about the behavior of estimators for linear problems than for nonlinear problems. The situation is even worse when the nonlinearities involve the highest derivatives occurring in the PDE being considered (strongly nonlinear PDEs). In this work we establish conditions under which those estimators, originally developed for linear problems, may be used for strongly nonlinear problems, and how that could be done. We also show that, under some suitable hypothesis, the estimators will be asymptotically exact, whenever they are asymptotically exact for linear problems. Those results allow anyone to use the knowledge about estimators developed for linear problems in order to build new reliable and robust estimators for nonlinear problems.     

Determining functionals for random partial differential equations

Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional random dynamical systems. In these applications the convergence condition of the trajectories of an infinite dimensional random dynamical system with respect to a finite set of linear functionals is assumed to be either in mean or exponential with respect to the convergence almost surely. In contrast to these ideas we introduce a convergence concept which is based on the convergence in probability. By this ansatz we get rid of the assumption of exponential convergence. In addition, setting the random terms to zero we obtain usual deterministic results.We apply our results to the 2D Navier-Stokes equations forced by a white noise.     

A posteriori estimates for the Stokes eigenvalue problem

We consider the Stokes eigenvalue problem. For the eigenvalues we derive both upper and lower a‐posteriori error bounds. The estimates are verified by numerical computations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Application of differential equations to synthesize a class of algorithms for numerical solution of a partial eigenvalue problem

We synthesize a class of recursive algorithms in the formation of which essential use is made of the properties of a certain type of differential equation and the qualitative analysis of it. These algorithms admit generalization to the case of nonsymmetric matrices. Translated fromAlgoritmy Upravleniya i Identifikatsii, pp. 104–117, 1997.     

Gradient bounds for anisotropic partial differential equations

We consider solutions in the whole of the space of a partial differential equation driven by the anisotropic Laplacian. We prove a pointwise energy bound and we derive from that some rigidity results.     

Analytical eigenvalue bounds for matrices arising when discretizing the self-adjoint second order elliptic partial differential equations

This paper shows how to derive analytical expressions for the eigenvalue bounds of matrices arising when using a fast method for separable finite difference equations for the numerical solution of the first three boundary value problems for the two-dimensional self-adjoint second order elliptic partial differential equation in a rectangle.     

On the general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations with singularities

A nonlinear self-adjoint eigenvalue problem for the general linear system of ordinary differential equations is examined on an unbounded interval. A method is proposed for the approximate reduction of this problem to the corresponding problem on a finite interval. Under the assumption that the initial data are monotone functions of the spectral parameter, a method is given for determining the number of eigenvalues lying on a prescribed interval of this parameter. No direct calculation of eigenvalues is required in this method.     

A posteriori error estimation for elliptic partial differential equations with small uncertainties

In this article, a finite element error analysis is performed on a class of linear and nonlinear elliptic problems with small uncertain input. Using a perturbation approach, the exact (random) solution is expanded up to a certain order with respect to a parameter that controls the amount of randomness in the input and discretized by finite elements. We start by studying a diffusion (linear) model problem with a random coefficient characterized via a finite number of random variables. The main focus of the article is the derivation of a priori and a posteriori error estimates of the error between the exact and approximate solution in various norms, including goal‐oriented error estimation. The analysis is then extended to a class of nonlinear problems. We finally illustrate the theoretical results through numerical examples, along with a comparison with the Stochastic Collocation method in terms of computational costs. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 175–212, 2016     

Minimax results for semicoercive functionals with application to differential equations

We give minimax theorems for some class of generalized convex and semicoercive functions. We define semicoercive saddle points and give sufficient conditions for functionals to have such critical points. Then we apply this method to show the existence of solutions for partial differential systems at resonance.      

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments.     

Box-spline–based approach to the formulation of numerical methods for partial differential equations

A multivariate box spline framework for the formulation of numerical methods for partial differential equations has been constructed. In particular, a fourth-order Galerkin method and a second-order collocation method were derived and applied to a test problem (classical Poisson equation on a square). The examples indicate that accuracy compares favorably with standard methods and the success of iterative schemes suggests an underlying stabilizing effect.     

A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals

For given 2n×2n matricesS ₁₃,S ₂₄ with rank(S ₁₃,S ₂₄)=2n we consider the eigenvalue problem:u′=A(x)u+B(x)v,v′=C ₁(x;λ)u-A ^T(x)v with

Nonlinear eigenvalue problem for a system of ordinary differential equations subject to a nonlocal condition

For a system of linear ordinary differential equations supplemented with a nonlocal condition specified by the Stieltjes integral, the problem of calculating the eigenvalues belonging to a given bounded domain in the complex plane is examined. It is assumed that the coefficient matrix of the system and the matrix function in the Stieltjes integral are analytic functions of the spectral parameter. A numerically stable method for solving this problem is proposed and justified. It is based on the use of an auxiliary boundary value problem and formulas of the argument principle type. The problem of calculating the corresponding eigenfunctions is also treated.     

On the number of an eigenvalue of a general nonlinear self-adjoint spectral problem for systems of ordinary differential equations

In the case of a general nonlinear self-adjoint spectral problem for systems of ordinary differential equations with boundary conditions independent of the spectral parameter, we introduce the notion of the number of an eigenvalue. Methods for the computation of the numbers of eigenvalues lying in a given range of the spectral parameter and for finding the eigenvalue with a given number, which were earlier suggested by the author for Hamiltonian systems, are generalized to the considered problem. We introduce the notion of an index of a problem for a general nontrivially solvable linear homogeneous self-adjoint boundary value problem.     

Upper spectral bounds and a posteriori error analysis of several mixed finite element approximations for the Stokes eigenvalue problem

This paper discusses conforming mixed finite element approximations for the Stokes eigenvalue problem. Firstly, several mixed finite element identities are proved. Based on these identities, the following new results are given: (1) It is proved that the numerical eigenvalues obtained by mini-element, P1-P1 element and Q1-Q1 element approximate the exact eigenvalues from above. (2) As for the P1-P1 , Q1-Q1 and Q1-P0 element eigenvalues, the asymptotically exact a posteriori error indicators are presented. (3) The reliable and efficient a posteriori error estimator proposed by Verfürth is applied to mini-element eigenfunctions. Finally, numerical experiments are carried out to verify the theoretical analysis.     

On the eigenvalue problem of a class of linear partial difference equations

In this paper, the eigenvalue problem of a class of linear partial difference equations is studied. The results concern the existence of eigenvalues, their character (real, positive), as well as the behavior of its eigenfunctions (positivity, oscillation). Moreover a theorem is given concerning the existence of a unique solution of an associated non-homogeneous partial difference equation. The results generalize previously known results for ordinary linear difference equations. The method used is a functional-analytic one, which transforms the eigenvalue problem for the difference equation into the equivalent problem of the eigenvalues of an operator defined on an abstract separable Hilbert space.