E-Locus - Institutional Repository of the University of Crete

Home Collections School/Department School of Sciences and Engineering Department of Mathematics and Applied Mathematics Post-graduate theses

Post-graduate theses

Current Record: 42 of 124

[Add to Basket]

Identifier

000442946

Title

Αριθμητική προσέγγιση της γραμμικής και ημιγραμμικής εξίσωσης του κύματος

Alternative Title

Numerical approximation of linear and semi-linear wave equation

Author

Φουκάκη, Ζαχαρένια

Thesis advisor

Ζουράρης, Γεώργιος

Abstract

Chapter 1: In the beginning of the current work an initial- and Dirichlet boundary- value problem for a semi-linear wave equation is formulated. Then using the energy method an a-priori bound of the solution, depending only on the problem data, is constructed. Subsequently the uniqueness of the solution to the semi-linear wave equation is proved combining the above bound and well-known results from the Sobolev spaces theory. Chapter 2: The assumptions on the finite element spaces and their approximation properties are introduced. The definition of the elliptic projection is given and the estimation of its ap¬proximation error in H 1(Ω) and L2(Q) norms are proved. In addition, the Taylor theorem with integral remainder, which is used to estimate the consistency error, is involved. The Chapter 2 is closed by discussing some distinct Gronwall's lemmas. Chapter 3: A numerical method is formulated in order to approximate the solution to the linear wave equation, which is obtained by discretizing the equation in time with the use of the Newmark method with parameter β e (1, 2 ] and in space with the use of the standard finite element method. The existence and uniqueness of the proposed numerical approximations are demonstrated, the consistency errors in time are estimated and, finally, a stability result is established. The convergence of the method is ensured by proving an optimal order error of the form 0(τ2 + hK) in the H 1(Ω) norm and of the form O(r2 + hK+1) in the £2(Ω) norm of the discrete first order time derivative of the error, where h and τ respectively denote the spatial and temporal discretization parameters. Chapter 4: In correspondence with the Chapter 3, a numerical method is formulated in order to approximate the solution to the semi-linear wave equation, which is obtained by discretizing the equation both in time with the use of a linearly implicit variant of the Newmark method with parameter β e (1, 2] and in space with the use of the finite element method. The existence and uniqueness of fully discrete approximations with the use of energy method are demonstrated and also the consistency errors in time are estimated. In addition, an a-priori barrier is constructed for the numerical approximations which depend only on the choice of norms and the regularity of the solution. The above barrier is independent of the spatial and temporal discretization parameters. In conclusion, convergence errors of order 0(τ2 + hK) in the norm H 1(Ω) and of order 0(τ2 + hK+1) with respect to the norm £2(Ω) of the discrete first order time derivative are demonstrated with the use of elements from theory of Sobolev spaces and the already proved a-priori barriers for the numerical approximated solutions.

Language

Greek

Subject

Consistency estimation errors

Convergence of method

Discrete gronwall's lemma

Elliptical projection

Numerical method

Optimal order of estimation error

Stability result

Ύπαρξη και μοναδικότητα αριθμητικής προσέγγισης

Αριθμητική μέθοδος