Nonstandard Discretization Strategies In Isogeometric Analysis for Partial Differential Equations

DI Stephen Edward Moore

April 28, 2017, 2 p.m. S3 057


\emph{Isogeometric Analysis} (IgA), based on B-spline and Non-Uniform Rational B-Spline (NURBS), is a numerical method proposed in 2005 by
Thomas Hughes, John Cottrell and Yuri Bazilevs to approximate
solutions of
partial differential
equations (PDEs). IgA uses the same class of basis functions for both representing the geometry of the computational domain and approximating the
solution of problems modeled by PDEs.
NURBS basis functions are the main underlying tools in most industrial and engineering design processes.
The special properties associated with NURBS including the ease of constructing
basis functions with
higher smoothness
and special refinement strategies makes them a very suitable choice in many real life applications.
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In many engineering or practical applications, the computational domains
cannot be represented by a single NURBS geometry mapping, and thus must be decomposed into several subdomains.
These subdomains are referred to as \emph{patches}
in IgA. In this regards, we developed a multipatch \emph{discontinuous Galerkin Isogeometric Analysis} (dG-IgA)
of PDEs given in volumetric computational domains as well as on closed and open surfaces. We present the numerical analysis of the dG-IgA schemes
proposed, and show \emph{a priori} error estimates for geometrically matching subdomains with \emph{hanging nodes} on the interface, i.e., non-matching meshes
are allowed.
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Many realistic applications involve complex domains with non-smooth boundary parts,
changing boundary conditions,
non-smooth coefficients arising from material interfaces etc.
It is well known that standard numerical schemes on uniform meshes
do not yield optimal convergence rate. This is due to the reduced regularity
of the solution in the vicinity
where the singularities occur. We therefore develop and analyze a \emph{graded mesh} technology in isogeometric analysis which leads to the desired
and expected optimal convergence rate. The IgA \emph{mesh grading} uses a priori information of the behavior of the solution around the points, where
the singularity occurs, and create an appropriate mesh sequence yielding
the same convergence rate as in the smooth case.
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Finally, we consider linear parabolic initial-boundary value problems. There are several well-known classical time-stepping
schemes for solving parabolic evolution problems like implicit and explicit Runge-Kutta methods. In this thesis, we present
a \emph{space-time }
IgA
of parabolic evolution problems
as an alternative approach to the numerical solution of
time-dependent PDE problems.
By using \emph{time-upwind} test functions, we derive a
stable space-time IgA
and space-time dG-IgA
scheme that combines very well with parallel solvers.
We consider
both fixed and moving
spatial computational domains.
\emph{A priori} error estimates with respect to some discrete energy norm
are presented. Our numerical experiments confirm these theoretical
results.