Energy Error Estimates Research Articles

Approximation of multidimensional Stefan-like problems via hyperbolic relaxation

An effective time delay in the governing equations, as proposed in [8], is further examined here. The resulting hyperbolic Stefan problem is mainly viewed as a perturbation of the traditional one. Asymptotic energy error estimates are derived for the three physical unknowns (heat flux, temperature and enthalpy) in accordance with various regularity assumptions on the initial data.

An Efficient Linear Scheme to Approximate Parabolic Free Boundary Problems: Error Estimates and Implementation

This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of $O({h^{1/2}})$ is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown.

An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation

This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of O ( h 1 / 2 ) O({h^{1/2}}) is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown.

The combined use of a nonlinear chernoff formula with a regularization procedure for two-phase stefan problems

The approximation of two-phase Stefan problems in 2-D by a nonlinear Chernoff formula combined with a regularization procedure is analyzed. The first technique allows the associated strongly nonlinear parabolic P.D.E. to be approximated by a sequence of linear elliptic problems. In addition, non-degeneracy properties can be properly exploited through the use of a smoothing process. A fully discrete scheme involving piecewise linear and constant finite elements is proposed. Energy error estimates are proven for both physical variables, namely enthalpy and temperature. These rates of convergence improve previous results.

Energy error estimates for a linear scheme to approximate nonlinear parabolic problems

On considere un schema de discretisation suivant la variable temporelle, issu de la theorie des semi-groupes de contractions non lineaires pour l'approximation des problemes paraboliques fortement non lineaires (problemes de Stefan multidimensionnels, equations de milieux poreux). On etablit des estimations d'erreur. On utilise une technique variationnelle

An Interior Penalty Finite Element Method with Discontinuous Elements

A new semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed. The test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes with interelement continuity enforced approximately by means of penalties. Optimal order error estimates in energy and $L^2$-norms are stated in terms of locally expressed quantities. They are proved first for a model problem and then in general.

Spectral and pseudo-spectral methods for parabolic problems with non periodic boundary conditions

The advection-diffusion equation is approximated by Chebyshev and Legendre spectral and pseudo-spectral methods. Stability results in the energy norm and error estimates in terms of the discretization parameter and of the regularity of the solution in weighted Sobolev norms are presented.

Energy inequalities and error estimates for torsion of elastic shells of revolution

This paper is concerned with the problem of axisymmetric torsion by terminal loads of elastic shells of revolution. Such shells are considered here to be three-dimensional bodies occupying a region of space which includes all points whose distances from a given surface - the midsurface - do not exceed h/2, where h is the shell thickness. The analysis is based on the classical linear theory of elasticity for homogeneous, isotropic materials, and it may be regarded as an extension of the work described in [1] and [2]. The purpose of the present paper is to assess the quality of an approximate solution of the thin shell problem. The case of axisymmetric torsion of thin shells of revolution is perhaps tile simplest one for this purpose, since simple approximate solutions - constructed from two-dimensional shell theories or otherwise - are known [3]. Our results provide estimates, based on the three-dimensional theory of elasticity, for the error involved in a stress analysis in which the 'exact' solution is replaced by an approximate one.