Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow

Summary. We consider the finite element approximation of a non-Newtonian flow, where the viscosity obeys a general law including the Carreau or power law. For sufficiently regular solutions we prove energy type error bounds for the velocity and pressure. These bounds improve on existing results in the literature. A key step in the analysis is to prove abstract error bounds initially in a quasi-norm, which naturally arises in degenerate problems of this type. Received May 25, 1993 / Revised version received January 11, 1994     

A priori error estimates for the finite element approximation of an obstacle problem

The purpose of this paper is to measure, with explicit constants as small as possible, a priori error bounds for approximation by picewise polynomials. These constants play an important role in the numerical verification method of solutions for obstacle problems by using finite element methods.     

Optimal error estimates of mixed finite element methods for a fourth-order nonlinear elliptic problem

A mixed finite element method is developed for a nonlinear fourth-order elliptic problem. Optimal L² error estimates are proved by using a special interpolation operator on the standard tensor-product finite elements of order k?1. Then two iterative schemes are presented and proved to keep the same optimal error estimates. Three numerical examples are provided to support the theoretical analysis.     

Error analysis of finite element approximation of a stefan problem with nonlinear free boundary condition

By applying the Landau-type transformation, we transform a Stefan problem with nonlinear free boundary condition into a system consisting of a parabolic equation and the ordinary differential equations. Fully discrete finite element method is developed to approximate the solution of a system of a parabolic equation and the ordinary differential equations. We derive optimal orders of convergence of fully discrete approximations inL2, H¹ and H² normed spaces.     

Explicit error bounds in a conforming finite element method

The goal of this paper is to define a procedure for bounding the error in a conforming finite element method. The new point is that this upper bound is fully explicit and can be computed locally. Numerical tests prove the efficiency of the method. It is presented here for the case of the Poisson equation and a first order finite element approximation.

     

Robust a posteriori error estimation for the nonconforming Fortin-Soulie finite element approximation

We obtain a computable a posteriori error bound on the broken energy norm of the error in the Fortin-Soulie finite element approximation of a linear second order elliptic problem with variable permeability. This bound is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms. The estimator is completely free of unknown constants and provides a guaranteed numerical bound on the error.

     

Half-line solutions of a nonlinear heat conduction problem

We solve a half-line problem for a nonlinear diffusion equation with a given time-dependent thermal conductivity at the origin. The problem reduces to a linear Volterra integral equation, which is solvable by Picard’s process of successive approximations. We analyze some explicit examples numerically. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 58–65, July, 2007.     

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.     

Error bounds for the finite element approximation of a degenerate quasilinear parabolic variational inequality

In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ? ^d , withd=1 or 2. GivenT>0,p ∈ (1, ∞),f andu ₀; findu ∈K, whereK is a closed convex subset of the Sobolev spaceW ₀ ^1,p (Ω), such that for anyv∈K $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular.     

An inverse heat conduction problem with a nonlinear source term

In this paper, the nonlinear problem of an inhomogeneous heat equation with linear boundary conditions will be considered. The surface heat flux history of a heated conducting body will be identified. The approach of the proposed method is to approximate the unknown function using linear polynomial pieces which are determined consecutively from the solution of the minimization problem on the basis of overspecified data. Some numerical examples will also be presented.     

On the finite element approximation of a cascade flow problem

Summary The finite element analysis of a cascade flow problem with a given velocity circulation round profiles is presented. The nonlinear problem for the stream function with nonstandard boundary conditions is discretized by conforming linear triangular elements. We deal with the properties of the discrete problem and study the convergence of the method both for polygonal and nonpolygonal domains, including the effect of numerical integration.     

Quadratic finite element approximation of the Signorini problem

Applying high order finite elements to unilateral contact variational inequalities may provide more accurate computed solutions, compared with linear finite elements. Up to now, there was no significant progress in the mathematical study of their performances. The main question is involved with the modeling of the nonpenetration Signorini condition on the discrete solution along the contact region. In this work we describe two nonconforming quadratic finite element approximations of the Poisson-Signorini problem, responding to the crucial practical concern of easy implementation, and we present the numerical analysis of their efficiency. By means of Falk's Lemma we prove optimal and quasi-optimal convergence rates according to the regularity of the exact solution.

     

A framework for obtaining guaranteed error bounds for finite element approximations

We give an overview of our recent progress in developing a framework for the derivation of fully computable guaranteed posteriori error bounds for finite element approximation including conforming, non-conforming, mixed and discontinuous finite element schemes. Whilst the details of the actual estimator are rather different for each particular scheme, there is nonetheless a common underlying structure at work in all cases. We aim to illustrate this structure by treating conforming, non-conforming and discontinuous finite element schemes in a single framework. In taking a rather general viewpoint, some of the finer details of the analysis that rely on the specific properties of each particular scheme are obscured but, in return, we hope to allow the reader to ‘see the wood despite the trees’.     

Mixed finite element approximation of a degenerate elliptic problem

Summary. We present a mixed finite element approximation of an elliptic problem with degenerate coefficients, arising in the study of the electromagnetic field in a resonant structure with cylindrical symmetry. Optimal error bounds are derived. Received May 4, 1994 / Revised version received September 27, 1994     

Finite element approximation of a problem with a nonlinear Newton boundary condition

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous problem is established with the aid of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numerical integration for the computation of nonlinear boundary integrals. The solvability of the discrete finite element problem is proved and the convergence of the approximate solutions to the exact one is analysed. Received April 15, 1996 / Revised version received November 22, 1996     

Error analysis for a finite element approximation of a thermoviscoelastic contact problem

In this article, a finite element approximation, based on a variational inequality, to the solution of a one-dimensional quasi-static Signorini contact problem in linear thermoviscoelasticity is proposed. Stability and error estimates are obtained.     

Uniqueness of moving boundary for a heat conduction problem with nonlinear interface conditions

In this paper, based on the maximum principle and the unique continuation theorem, we present a uniqueness result for a moving boundary of a heat problem in a multilayer medium with nonlinear interface conditions.     

EQ 1 rot nonconforming finite element method for nonlinear dual phase lagging heat conduction equations

EQ rot 1 nonconforming finite element approximation to a class of nonlinear dual phase lagging heat conduction equations is discussed for semi-discrete and fully-discrete schemes. By use of a special property, that is, the consistency error of this element is of order O(h2 ) one order higher than its interpolation error O(h), the superclose results of order O(h2 ) in broken H1 -norm are obtained. At the same time, the global superconvergence in broken H1 -norm is deduced by interpolation postprocessing technique. Moreover, the extrapolation result with order O(h4 ) is derived by constructing a new interpolation postprocessing operator and extrapolation scheme based on the known asymptotic expansion formulas of EQ rot 1 element. Finally, optimal error estimate is gained for a proposed fully-discrete scheme by different approaches from the previous literature.