Title page for ETD etd-0821102-125038

Document Type thesis

Author Name Tu, Xuemin

URN etd-0821102-125038

Title Enhanced Singular Function Mortar Finite Element Methods

Degree MS

Department Mathematical Sciences

Advisors
Marcus Sarkis, Advisor

Keywords
Mortar Finite Element
Singular Function

Date of Presentation/Defense 2002-04-30

Availability unrestricted

Abstract
It is well known that singularities occur when solving elliptic
value problems with non-convex domains or when some part of the data or the
coefficients of the PDE are not smooth. Such problems and correspondent
singularities
often arise in practice, for instance, in fracture mechanics, in the
material science with heterogeneities, or when dealing with
mixed boundary conditions. A great deal is known about
the nature of the singularities, which arise in some of these problems.
In this thesis,
we consider the scalar transmission problems with straight interfaces
and with cross points across coefficients and possibly on a
non-convex region ($L$-shaped domain).
It is known that only $H^{1+ au}$ ($0 < au< 1$) regularity on
the solution
is obtained and therefore the use of finite element method with
the piecewise linear continuous function space does not give optimal accuracy.
In this thesis, we introduce a new algorithm which are second order
accurate on the (weighted) $L_2$, first order accurate on the
(weighted) $H_1$ norm and second order accurate for the Stress Intensive
Factor (SIF). The new methods take advantage of Mortar techniques.
The main feature
of the proposed algorithms is that we use primal singular functions
{it without} cutting-off functions. The old algorithms use cutting-off
functions as a tool of satisfying boundary conditions. In algorithms proposed
in this thesis, use instead Mortar finite element technique
to match the boundary and interfaces conditions. In this thesis,
we also consider non-matching meshes sizes for
different coefficients. We note that a new Mortar Lagrange multiplier
is required in order to obtain optimal consistence errors for transmission
problems. The proposed algorithms are very appealing over other
methods because they are very accurate, do not
require complicated numerical quadratures or interpolations, it is
simple to design PCGs, and it can be generalized to other PDEs and
to higher order methods.

Files
xuemintu.pdf

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Document Type	thesis
Author Name	Tu, Xuemin
URN	etd-0821102-125038
Title	Enhanced Singular Function Mortar Finite Element Methods
Degree	MS
Department	Mathematical Sciences
Advisors	Marcus Sarkis, Advisor
Keywords	Mortar Finite Element Singular Function
Date of Presentation/Defense	2002-04-30
Availability	unrestricted
Abstract It is well known that singularities occur when solving elliptic value problems with non-convex domains or when some part of the data or the coefficients of the PDE are not smooth. Such problems and correspondent singularities often arise in practice, for instance, in fracture mechanics, in the material science with heterogeneities, or when dealing with mixed boundary conditions. A great deal is known about the nature of the singularities, which arise in some of these problems. In this thesis, we consider the scalar transmission problems with straight interfaces and with cross points across coefficients and possibly on a non-convex region ($L$-shaped domain). It is known that only $H^{1+ au}$ ($0 < au< 1$) regularity on the solution is obtained and therefore the use of finite element method with the piecewise linear continuous function space does not give optimal accuracy. In this thesis, we introduce a new algorithm which are second order accurate on the (weighted) $L_2$, first order accurate on the (weighted) $H_1$ norm and second order accurate for the Stress Intensive Factor (SIF). The new methods take advantage of Mortar techniques. The main feature of the proposed algorithms is that we use primal singular functions {it without} cutting-off functions. The old algorithms use cutting-off functions as a tool of satisfying boundary conditions. In algorithms proposed in this thesis, use instead Mortar finite element technique to match the boundary and interfaces conditions. In this thesis, we also consider non-matching meshes sizes for different coefficients. We note that a new Mortar Lagrange multiplier is required in order to obtain optimal consistence errors for transmission problems. The proposed algorithms are very appealing over other methods because they are very accurate, do not require complicated numerical quadratures or interpolations, it is simple to design PCGs, and it can be generalized to other PDEs and to higher order methods.
Files	xuemintu.pdf