The purpose of this thesis is to extend the range of application of the method fundamental solutions (MFS) to solve direct and inverse geometric problems associated with two- or three-dimensional Helmholtz-type equations. Inverse problems have become more and more important in various fields of science and technology, and have certainly been one of the fastest growing areas in applied mathematics over the last three decades. However, as inverse geometric problems typically lead to mathematical models which are ill-posed, their solutions are unstable under data perturbations and classical numerical techniques fail to provide accurate and stable solutions.

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This thesis studies the phenomena of rate-induced bifurcations. Externally forced systems may have a critical rate, above which they undergo some sort of destabilisation, and move away suddenly to a new state. Mathematically, the phenomenon is a non-autonomous instability. We present a framework in which rate-induced bifurcations can be studied. This is based on geometric singular perturbation theory which is derived from Fenichel’s Theorem. In particular we make use of folded singularities and canard trajectories, which are modern concepts from geometric singular perturbation theory. We concentrate on systems with multiple time-scales where the mechanism for a rate-induced bifurcation is not obvious. So much so, that once a multiple time-scale system has undergone a rate-induced bifurcation, the instability threshold which separates initial states that destabilise from those that adiabatically follow a changing stable state is described as non-obvious. We study in detail the complicated non-obvious instability threshold that arises near a folded saddle-node (type I) singularity. In particular, we show how the complicated threshold structure depends on two parameters – the ratio of time-scales and the the folded singularity bifurcation parameter. In contrast, we also show single time-scale systems where the rate-induced bifurcation is caused by a large perturbation in the boundary of the basin of attraction for the stable state.

The main result proved in this thesis is an existence theorem for asymptotically conical Ricci-flat Kahler metrics on C2 with cone singularities along a smooth complex curve. These metrics are expected to arise as blow-up limits of non-collapsed sequences of Kahler-Einstein metrics with cone singularities.

The thesis studies the approximation properties of splines with maximum smoothness. We are interested in the behaviour of the approximation as the degree of the spline increases (so does its smoothness). By studying B-spline interpolation, we obtain error estimates measured in the semi-norm that are explicit in terms of mesh size, degree and smoothness. This new result also gives a higher approximation order than existing estimations. With the results, we investigate the B-spline finite element approximation with k-refinement, which is a strategy of improving the accuracy by increasing the degree and smoothness. The problem is studied in the setting of heat equations and wave equations. We give B-spline FEM schemes for the problems, and obtain error estimates. Moreover, by proving a Markov-type inequality for splines, where an exact constant is derived, we deduce how the stability of the scheme behaves with the k-refinements. We also improve the efficiency of the schemes for problems with periodic boundary conditions by applying the fast Fourier transform. The thesis also focuses on developing algorithms for efficiently evaluating the element system matrices in finite element methods with Berstein-Bâezier splines as shape functions, where the splines are of arbitrary order and defined on quadrilaterals and hexahedrons. The algorithms achieve the optimal complexity by making use of the sum factorial procedure. We test the algorithms in C++ implementation, and the numerical results illustrate that the optimal cost and expected accuracy are achieved.

Based on Guillemin’s work on gauged Lagrangian distributions, we will introduce the notion of a gauged poly-log-homogeneous distribution as an approach to ζ-functions for a class of Fourier Integral Operators which includes cases of amplitudes with asymptotic expansion Σk∈N amk where each amk is log-homogeneous with degree of homogeneity mk but violating R(mk) → −∞. We will calculate the Laurent expansion for the ζ-function and give formulae for the coefficients in terms of the phase function and amplitude, as well as investigate generalizations to the Kontsevich-Vishik trace. Using stationary phase approximation, series representations for the Laurent coefficients and values of ζ-functions will be stated explicitly, and the kernel singularity structure will be studied. This will yield algebras of Fourier Integral Operators which purely consist of Hilbert-Schmidt operators and whose ζ-functions are entire, as well as algebras in which the generalized Kontsevich- Vishik trace is form-equivalent to the pseudo-differential operator case. Additionally, we will introduce an approximation method (mollification) for ζ-functions of Fourier Integral Operators whose amplitudes are poly-log-homogeneous at zero by ζ-functions of Fourier Integral Operators with “regular” amplitudes. In part II, we will study Bochner-, Lebesgue-, and Pettis integration in algebras of Fourier Integral Operators. The integration theory will extend the notion of parameter dependent Fourier Integral Operators and is compatible with the Atiyah-Jänich index bundle as well as the ζ-function calculus developed in part I. Furthermore, it allows one to emulate calculations using holomorphic functional calculus in algebras without functional calculus, and to consider measurable families of Fourier Integral Operators as they appear, for instance, in heat- and wave-traces of manifolds whose metrics are subject to random (possibly singular) perturbations.

The main object to study in this thesis is the so-called damped and driven Navier-Stokes equations. These equations differ from the classical Navier-Stokes system by the presence of the extra damping term which is greater than zero, which is often referred to as the Ekman damping term and models the bottom friction in two-dimensional oceanic models.

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In this work, we make a detailed study of the Fourier coefficients of cuspidal Siegel modular forms of degree 2. We derive a very general relation between the Fourier coefficients that extends previous work in this direction by Andrianov, Kowalski-Saha-Tsimerman and others. The basis for our relation is the dependence between values of global Bessel periods and averages of Fourier coefficients. Consequently our relation applies also to Bessel periods of more general automorphic forms on GSp4(A). We use our relation to prove that cuspidal Siegel modular forms associated to P-CAP representations (Saito-Kurokawa lifts with level) satisfy the so-called Maass relations. This is the first result of this kind for Siegel modular forms with respect to general congruence subgroups. Another important corollary of our work is the existence of non-zero Fourier coefficients of the simplest form possible (often fundamental or primitive) for a wide family of cuspidal Siegel modular forms of degree 2. Finally, using classical methods, we are able to prove that paramodular newforms of square-free level have infinitely many non-zero fundamental Fourier coefficients. This result extends previous work by Saha in the full-level case, and is especially interesting because of the paramodular conjecture connecting paramodular newforms of weight 2 and rational abelian surfaces.