Publication

Adaptive algorithms for two fluids flows with anisotropic finite elements and order two time discretizations

Samuel Dubuis
2020
EPFL thesis
Abstract

This thesis is devoted to the derivation of a posteriori error estimates for the numerical approximation of fluids flows separated by a free surface. Based on these estimates, error indicators are introduced and adaptive algorithms are proposed to solve the problem with accuracy and low computational costs. We focus on numerical methods that are combinations of anisotropic finite elements and second order methods to advance in time.

We split the technical difficulties in the derivation of the error estimates by first studying independent PDEs, and in a second time by gathering the different results to analyse the complete system of equations composed with these latter. The a posteriori error analysis for the approximation of these PDEs will be addressed in a particular and devoted chapter. The last chapter is dedicated to the study of the system describing two fluids flows.

In each chapter, we focus on two main objectives. The first is a theoretical analysis and the derivation of error estimates, the second is the description and the implementation of an algorithm to adapt meshes and time steps. Finally, numerical experiments are performed to demonstrate the efficiency of the procedure.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related concepts (33)
Numerical methods for ordinary differential equations
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.
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.
Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.
Show more
Related publications (158)

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

Paride Passelli

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 ...
EPFL2024

Anisotropic Adaptive Finite Elements for a p-Laplacian Problem

Marco Picasso, Paride Passelli

The p-Laplacian problem -del & sdot; ((mu + |del u|(p-2))del u) = f is considered, where mu is a given positive number. An anisotropic a posteriori residual-based error estimator is presented. The error estimator is shown to be equivalent, up to higher ord ...
Walter De Gruyter Gmbh2024

Tensor approximation of the self-diffusion matrix of tagged particle processes

Christoph Max Strössner

The objective of this paper is to investigate a new numerical method for the approximation of the self-diffusion matrix of a tagged particle process defined on a grid. While standard numerical methods make use of long-time averages of empirical means of de ...
ACADEMIC PRESS INC ELSEVIER SCIENCE2023
Show more
Related MOOCs (32)
Warm-up for EPFL
Warmup EPFL est destiné aux nouvelles étudiantes et étudiants de l'EPFL.
Matlab & octave for beginners
Premiers pas dans MATLAB et Octave avec un regard vers le calcul scientifique
Matlab & octave for beginners
Premiers pas dans MATLAB et Octave avec un regard vers le calcul scientifique
Show more

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.