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Publication# Adaptive Finite Elements with Large Aspect Ratio. Application to Aluminium Electrolysis

Abstract

The goal of this work is to use anisotropic adaptive finite elements for the numerical simulation of aluminium electrolysis. The anisotropic adaptive criteria are based on a posteriori error estimates derived for simplified problems. First, we consider an elliptic problem with smooth, but strongly varying coefficient. We then study the steady Stokes equation and finally the nonlinear p-Laplace problem. In particular, we focus on continuous piecewise linear finite elements with possibly large aspect ratio. Whenever possible, error estimates are proven to be equivalent to the numerical error with constants independent from the mesh aspect ratio. Numerical experiments confirming these predictions are presented.Adaptive strategies based on the derived error estimates are proposed. Numerical experiments show sharpness of the estimates on adapted meshes. For a given accuracy the computational time is reduced. Finally, numerical simulations of the aluminium electrolysis process, using adaptive meshes, are considered. The fluid-flow problem without and with gas is studied. Numerical results confirming a reduction of the computational time are presented.

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Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Computer simulation

Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determined by comparing their results to the real-world outcomes they aim to predict. Computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics (computational physics), astrophysics, climatology, chemistry, biology and manufacturing, as well as human systems in economics, psychology, social science, health care and engineering.

Approximation error

The approximation error in a data value is the discrepancy between an exact value and some approximation to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value). An approximation error can occur for a variety of reasons, among them a computing machine precision or measurement error (e.g. the length of a piece of paper is 4.53 cm but the ruler only allows you to estimate it to the nearest 0.

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