Κατεύθυνση Εφαρμοσμένα Μαθηματικά
Library of the School of Science

2017-02-28

2016

Svolos Lampros

Δουγαλής Βασίλειος Καθηγητής Τμημ. Μαθηματικών ΕΚΠΑ, Δρακόπουλος Μιχαήλ Επίκ. Καθηγητής Τμημ. Μαθηματικών ΕΚΠΑ, Νοτάρης Σωτήριος Αναπλ. Καθηγητής Τμημ. Μαθηματικών ΕΚΠΑ

Μέθοδοι Galerkin - πεπερασμένων στοιχείων για υπερβολικές Μ.Δ.Ε. πρώτης τάξης

Greek

Μέθοδοι Galerkin – πεπερασμένων στοιχείων για υπερβολικές ΜΔΕ πρώτης τάξης
Finite Element Methods for first-order hyperbolic PDEs

The purpose of this master thesis is to study finite element methods (Galerkin methods) for first-order hyperbolic PDEs. Specifically, approximate methods will be used to solve the transport equation.
In the second chapter, we will provide some theoretical results about the linear transport equation
ut + α ux = f(x,t)
In particular, the method of characteristics is employed to determine the boundary and initial conditions with which the problem will be well-defined. This procedure leads us to find the problem that will be the model for implementation of numerical methods which will be presented in the following chapters.
The third chapter examines the standard Galerkin methods. Finite element methods will be presented by using splines in which the space of trial functions is the same with the space of test functions. To formulate the error estimates for the aforementioned methods we derive inverse inequalities for finite-dimensional spaces such as the space of piecewise linear functions or the space of cubic splines. Furthermore, theorems for the error estimates of continuous time Galerkin approximation are formulated and are proved. For example, if we have uniform mesh we get higher rate of convergence than non-uniform mesh. This is super-convergence which is confirmed by conducting numerical experiments for different types of discrete time Galerkin methods. Specifically, the numerical schemes that we apply to conduct these experiments combine the following techniques. The trapezoidal rule or the 4th order Runge-Kutta is used to advance the equation in time while the space of piecewise linear functions or the space of cubic splines is used to discretize the equation in space. All these combinations are presented in this chapter.
Chapter four concentrates on the Baker’s method which published in 1975 and give approximate solutions for first-order hyperbolic differential equations. The method is based on nonstandard variational formulation of transport equation. The Galerkin approximation is obtained by the use of specially chosen spaces of trial functions and test functions in a weak formulation of the boundary value problem. Specifically, the above spaces are subspaces of L2 and H1 respectively and are chosen to be compatible with the aforementioned variational formulation. At first, the necessary definitions of function spaces are provided because these spaces will be used in the proofs of error estimates. The solution is approximated in a space of discontinuous piecewise polynomial functions of degree r−1, r ≥1 and the test functions are defined to create a pair of spaces which is compatible with the variational formulation. We will present the bases of these spaces and we can prove that they have the same dimension. Some important theorems subsequently will be proved such as a projection theorem. The error estimates for the following schemes can be proved.
• 1st Scheme: (semi-discrete) The approximation is continuous in the time variable with optimal L2 error estimates of O(hr)
• 2nd Scheme: (discrete time Method) A Crank-Nicolson type time discretization yields approximations discrete in time with optimal L2 error estimates of O(hr+τ2), where τ denotes the discrete time step. The method is unconditionally convergent and stable.
Finally, these inequalities for error estimates are confirmed by running some numerical experiments. The behavior of Baker’s method in problems with discontinuities (e.g. jumps) is examined.

Science

Mathematics

Finite Element Methods
Hyperbolic PDEs
Transport Equation
Standard Galerkin Methods
G. Baker’s Method
Nonstandard Galerkin Methods

No

0

Yes

8

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https://pergamos.lib.uoa.gr/uoa/dl/object/1327888