dc.contributor.author |
ADEWOLE, MATTHEW OLAYIWOLA, |
|
dc.date.accessioned |
2019-03-21T10:23:21Z |
|
dc.date.available |
2019-03-21T10:23:21Z |
|
dc.date.issued |
2017-05 |
|
dc.identifier.uri |
http://hdl.handle.net/123456789/297 |
|
dc.description.abstract |
Nonlinear parabolic interface problems are frequently encountered in the modelling
of physical processes which involved two or more materials with di erent properties.
Research had focused largely on solving linear parabolic interface problems
with the use of Finite Element Method (FEM). However, Spectral Element Method
(SEM) for approximating nonlinear parabolic interface problems is scarce in literature.
This work was therefore designed to give a theoretical framework for the
convergence rates of nite and spectral element solutions of a nonlinear parabolic
interface problem under certain regularity assumptions on the input data.
A nonlinear parabolic interface problem of the form
ut r (a(x; u)ru) = f(x; u) in
(0; T]
with initial and boundary conditions
u(x; 0) = u0(x) ; u(x; t) = 0 on @
[0; T]
and interface conditions
[u] = 0;
a(x; u)
@u
@n
= g(x; t)
was considered on a convex polygonal domain
2 R2 with the assumption that
the unknown function u(x; t) is of low regularity across the interface, where f :
R ! R, a :
R ! R are given functions and g : [0; T] ! H2() \ H1=2()
is the interface function. Galerkin weak formulation was used and the solution
domain was discretised into quasi-uniform triangular elements after which the
unknown function was approximated by piecewise linear functions on the nite
elements. The time discretisation was based on Backward Di erence Schemes
(BDS). The implementation of this was based on predictor-corrector method due
to the presence of nonlinear terms. A four-step linearised FEM-BDS was proposed
and analysed to ease the computational stress and improve on the accuracy of the
ii
time-discretisation. On spectral elements, the formulation was based on Legendre
polynomials evaluated at Gauss-Lobatto-Legendre points. The integrals involved
were evaluated by numerical quadrature. The linear theories of interface and noninterface
problems as well as Sobolev imbedding inequalities were used to obtain
the a priori and the error estimates. Other tools used to obtain the error estimates
were approximation properties of linear interpolation operators and projection
operators.
The a priori estimates of the weak solution were obtained with low regularity
assumption on the solution across the interface, and almost optimal convergence
rates of O
h
1 + 1
j log hj
1=2
and O
h2
1 + 1
j log hj
in the L2(0; T;H1(
)) and
L2(0; T; L2(
)) norms respectively were established for the spatially discrete scheme.
Almost optimal convergence rates of O
k + h
1 + 1
j log hj
and
O
k + h2
1 + 1
j log hj
in the L2(0; T;H1(
)) and L2(0; T; L2(
)) norms were obtained
for the fully discrete scheme based on the backward Euler scheme, respectively
for small mesh size h and time step k. Similar error estimates were obtained
for two-step implicit scheme and four-step linearised FEM-BDS. The solution by
SEM was found to converge spectrally in the L2(0; T; L2(
))-norm as the degree
of the Legendre polynomial increases.
Convergence rates of almost optimal order in the L2(0; T;H1(
)) and
L2(0; T; L2(
)) norms for nite element approximation of a nonlinear parabolic
interface problem were established when the integrals involved were evaluated by
numerical quadrature.
Keywords: Finite element, Spectral element, Nonlinear parabolic problem
Word count: 438 |
en_US |
dc.language.iso |
en |
en_US |
dc.subject |
Finite element, Spectral element, Nonlinear parabolic problem |
en_US |
dc.title |
GALERKIN APPROXIMATION OF A NONLINEAR PARABOLIC INTERFACE PROBLEM ON FINITE AND SPECTRAL ELEMENTS |
en_US |
dc.type |
Thesis |
en_US |