Prescribed Error Tolerance Research Articles

Adaptive error analysis in seepage problems

AbstractNumerical experiments in adapting variations of a computationally simple error estimator (the Zienkiewicz‐Zhu estimator) to an existing finite element code are shown. The error estimator used allows both overall and local errors to be estimated. From the local estimates of error, refinements of the mesh are calculated to reach a prescribed error tolerance. These calculated refinements are used by a mesh refiner to produce a modified mesh which lowers the overall error to the prescribed value while keeping the mesh as crude as possible. The physical example on which these numerical experiments are performed is that of free surface flow through an earth dam with a toe drain. It is also shown how the problem formulation affects the error analysis and how the choice of computational scheme affects the mesh adaptation.

A Local Refinement Finite-Element Method for Two-Dimensional Parabolic Systems

We discuss an adaptive local refinement finite-element procedure for solving initial boundary value problems for vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The differential equations are discretized in space using piecewise bilinear finite-element approximations. An estimate of the spatial discretization error of the solution is obtained using piecewise cubic polynomials that employ nodal superconvergence to gain computational efficiency. The resulting system of ordinary differential equations for the finite-element solution and error estimate are integrated in time using existing software for stiff differential systems. The spatial error estimate is used to locally refine the finite-element mesh in order to satisfy a prescribed error tolerance. We discuss several aspects of the refinement algorithm and the dynamic tree data structure that is used to store the mesh, solution, and error estimate. A code that is based on our methods is applied to several examples in order to demonstrate the effectiveness of the error-estimation procedure and adaptive algorithms.

Fast Algorithm for Calculating Chemical Kinetics in Turbulent Reacting Flow

This paper addresses the need for a fast batch chemistry solver to perform the kinetics part of a split operator formulation of turbulent reacting flows, with special attention focused on the solution of the ordinary differential equations governing a homogeneous gas-phase chemical reaction. For this purpose, a two-part predictor-corrector algorithm which incorporates an exponentially fitted trapezoidal method was developed. The algorithm performs filtering of ill-posed initial conditions, automatic step-size selection, and automatic selection of Jacobi-Newton or Newton-Raphson iteration for convergence to achieve maximum computational efficiency while observing a prescribed error tolerance. The new algorithm, termed CREK1D (combustion reaction kinetics, one-dimensional), compared favorably with the code LSODE when tested on two representative problems drawn from combustion kinetics, and is faster than LSODE.

A moving-mesh finite element method with local refinement for parabolic partial differential equations

We discuss a moving-mesh finite element method for solving initial boundary value problems for vector systems of partial differential equations in one space dimension and time. The system is discretized using piecewise linear finite element approximations in space and a backward difference code for stiff ordinary differential systems in time. A spatial-error estimation is calculated using piecewise quadratic approximations that use the superconvergence properties of parabolic systems to gain computational efficiency. The spatial-error estimate is used to move and locally refine the finite element mesh in order to equidistribute a measure of the total spatial error and to satisfy a prescribed error tolerance. Ordinary differential equations for the spatial-error estimate and the mesh motion are integrated in time using the same backward difference software that is used to determine the numerical solution of the partial differential system. We present several details of an algorithm that may be used to develop a general-purpose finite element code for one-dimensional parabolic partial differential systems. The algorithm combines mesh motion and local refinement in a relatively efficient manner and attempts to eliminate problemdependent numerical parameters. A variety of examples that motivate our mesh-moving strategy and illustrate the performance of our algorithm are presented.