Abstract
This dissertation deals with different aspects of the shallow water equations (SWEs) on
the unit sphere in the three dimensional Euclidean space. Such equations arise by
vertically integrating the Navier--Stokes equations and the mass conservation equation.
They are a simplification of the equations describing various meteorological phenomena
such as tides in coastal areas. The SWEs are used as a primary test problem for new
efficient methods for real life hydrodynamical problems.
In this dissertation, firstly, we prove the existence of weak solutions for the SWEs in
consideration by the Galerkin method. We prove that the Galerkin approximations converge
to the solution of the original system.
When solving global meteorological problems, data are often collected by satellites.
Therefore it becomes inconvenient to use methods that require structured grids.
Consequently, the development of finite element methods that have less geometric
requirements for mesh generation becomes more and more popular.
A second contribution of this thesis is the design of a finite element method using
spherical splines in the case scattered data are involved in the modelling of the SWEs.
We implement our proposed method to the standard Williamson test set for the inviscid SWEs
with satellite data.
To generate more realistic flows it is important to include viscosity in the model.
As a third contribution, we derive {\it a priori} error estimates for this method using spherical splines,
when viscosity is included in both the momentum and continuity equations. We also calculate estimated
orders of convergence that support our theoretical results.
As is well known, the matrices arising when using finite element methods are often ill
conditioned. To overcome the ill conditioness, as a fourth contribution of the thesis,
we study two different types of preconditioners, namely the additive Schwarz and the
alternate triangular preconditioners. We calculate numerical bounds for the condition
numbers to verify the effectiveness of both preconditioners for the
spherical SWEs problem.