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Concept# Numerical analysis

Summary

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. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biolo

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This course offers an introduction to numerical methods for the solution of mathematical problems as: solution of systems of linear and non-linear equations, functions approximation, integration and differentiation and solution of differential equations.

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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.

In this thesis, we explore possible stabilisation methods for the reduce basis approximation of advection-diffusion problems, for which the advection term is dominating. The options we consider are mainly inspired by the Variational Multiscale method (VMS), which decomposes the solution of a variational problem into its coarse scale component, from a coarse scale space, and a fine scale component, from a fine scale space. Our stabilisation proposals are divided into three classes. The first one groups methods that rely on a stabilisation parameter. The second class uses VMS at the algebraic level to attempt stabilisation. Finally the third class is also inspired by VMS at the algebraic level, but with the additional constraint that the fine scale space is orthogonal to the coarse scale space. Numericals tests reported in this thesis show that the methods of the first class is not viable options as the best stabilisation parameter among those tested is the stabilisation parameter that is used at the high fidelity level. Although the stabilisation methods of the second class give accurate results when applied to stable problems, they were also dismissed by the numerical tests, as they did not improve the accuracy of the already stabilised problem. The third class also performs well when applied to stable problems. It has been shown in [7] one of those methods can improve accuracy. However in the current implementation, this result was not achieved here.

2016We numerically study the resistive method for the numerical approximation of elliptic PDEs. In particular we focus on the resistive method for weakly setting solution values in specific subdomains or interfaces in the domain.

2016