In this thesis, we study fractional differential equations with Hilfer derivative operators. Solutions are approximated using a newly developed Bernstein-splines approach and subsequently applied to the Van der Pol oscillator with fractional damping. Fractional derivatives generalize differentiation to the order α ∈ (0,∞), resulting in order α differential operators. The Hilfer-derivative of order α ∈ (0,1) and type β ∈ [0,1] is one of such operators and combines two of the most commonly used operators through parametrization: the Riemmann-Liouville-derivative (β = 0) and Caputo-derivative (β=1). The choice of β results in different kernel behavior of the operator, commonly yielding singular behavior of solutions of initial value problems (IVP's) and boundary value problems (BVP's) for β ∈ [0,1). Based on existing results for IVP's, we develop a new proof for existence and uniqueness of solutions to BVP's for Hilfer-fractional derivatives. To obtain solution approximations numerically for IVP's and BVP's, a Bernstein splines solution approach is developed and implemented, providing accurate convergence results for nonlinear IVP's and BVP's in an efficient vectorized approach. Implementation difficulties for the singular behavior of solutions for β ∈ [0,1) are successfully resolved through time-domain transformation approximation techniques. Finally, the methods are applied to numerically study the behavior of the fractionally damped Van der Pol oscillator, a nonlinear equation of interest in electrical engineering and control theory. We study the approximate limit cycle, of which behavior corresponds with existing analytical results for the Caputo-derivative (β=1). Convergence of solutions can also be obtained for Hilfer type values of β ∈ [0,1), appearing to be of little influence on the long-term limit cycle. Furthermore, when forcing is applied, we observe periodic, quasiperiodic and chaotic behavior.

Computational Fluid Dynamics (CFD) offers numerous benefits, notably the ability to study flows that are challenging or costly to investigate using experiments. A central challenge in CFD lies in simulating fluid flow around complex geometries. Additionally, the governing equations follow conservation laws. This thesis aims to establish the foundation for constructing Immersed divergence-conforming finite element spaces that address both challenges. To tackle the first issue, an immersed method is considered. Instead of generating a mesh that conforms to the object where the flow occurs, the approach involves placing the object within a predefined mesh. However, this introduces new challenges, the most significant of which is the ill-conditioning of the associated linear system. The condition number depends on the location of the immersed object and can lead to an extremely large condition number. The second challenge is resolved by discretizing the problem in a way that preserves essential topological and homological structures at a discrete level, utilizing the principles of finite element exterior calculus. Within this thesis, a structure-preserving subcomplex is developed for the de Rham complex in 1D, and its accuracy is validated through numerical experimentation. Optimal convergence is achieved, the discrete inf-sup test is satisfied, and the resulting linear system's conditioning remains unaffected by the placement of the immersed object. However, the constructed structure-preserving subcomplex for the de Rham complex in 2D, known as the immersed divergence-conforming finite element spaces, does not exhibit convergence. For future research, I recommend focusing on simpler vertical and skewed cuts before exploring more intricate immersed shapes. These simpler cuts involve straightforward choices for edge basis functions/1-form basis functions that are relevant. Additionally, if the outcomes remain unsatisfactory, considering a global approach instead of a local approach could be worthwhile.

Mimetic formulations, also known as structure-preserving methods, are numerical schemes that preserve fundamental properties of the continuous differential operators at a discrete level. Additionally, they are well-known for satisfying constraints such as conservation of mass or momentum.

In the present work, a Mimetic Spectral Element Method based on quadrilaterals is explored. As an introduction, the framework is first implemented and tested on the classical Poisson equation, the Hartmann Flow system and several eigenvalue problems for the Laplacian operator. Solutions are attained by direct/mixed formulations and the extension to multi-element approaches is dealt with using either gathering or connectivity matrices. Different boundary conditions and various geometries are utilized.

Afterwards, the Maxwell Eigenvalue problem for the electric field E with general material properties is tackled in an attempt to generate spurious-free solutions by incorporating the condition ∇·D = 0 into the discrete system. The formulation is further scrutinized on geometries with Betti number b₁ > 0 as to verify if the proposed scheme captures the physical zero eigenvalues.

In the end, a mixed formulation for the eigenproblem is proposed in which the curl-curl operator is separated. The approximation of the electrostatic field energy is then computed with this formulation and compared to the solution obtained with a direct method allowing to create an upper and a lower bound for this variable.

High-dimensional optimization problems with expensive and non-convex cost functions pose a significant challenge, as the non-convexity limits the viability of local optimization, where the results are sensitive to initial guesses and often only represent local minima. But as the number of expensive cost function evaluations required for a full exploration of the search space grows exponentially with the increasing number of dimensions, the use of standard global optimization algorithms is also not practical. To overcome this obstacle and to lower the dimensionality of the problem, the use of an autoencoder for model order reduction is proposed. In the resulting lower dimensional space, standard global optimization methods can then be utilized, as fewer cost functions evaluations are necessary. For problems with comparatively more expensive cost functions, this optimization includes the employment of a surrogate model, which reduces the necessary number of these computationally expensive evaluations further. This proposed method is then tested firstly on a number of benchmark functions, where it shows the ability to find global optima under certain conditions. Secondly, the proposed method is used to solve a compliance minimization problem, where it shows the ability to improve upon a large number of designs generated by local optimization.

Computational fluid dynamics is nowadays one of the pillars of modern aircraft design, just as impor­tant as experimental wind tunnel testing. Very ambitious goals in regards to performance, efficiency and sustainability are being asked of the aviation industry, the kind that warrant a virtual exploration of the edges of the flight envelope. High­ order has come to be regarded as a necessary ingredient to achieve breakthrough advances in this direction. So much so, in fact, that the major aircraft manufac­turers, governmental aerospace research agencies and top universities worldwide have been acting in coordination to increase the technology readiness level of these approaches, for the last ten years. In this work, I study in significant detail the one aspect that makes high-­order methods ideal candi­dates for enabling the use of more advanced turbulence models in industry: the cost­-efficiency of their discretization—they offer minimal amounts of dispersion and dissipation errors, for a given number of degrees of freedom. I consider three research objects: discontinuous Galerkin spectral method (DGSEM), flux reconstruction (FR) and a novel B-­spline-based discontinuous Galerkin formulation stabilized via algebraic flux correc­tion (DGIGA­-AFC), and characterize their order of accuracy, linear and nonlinear stability characteristics, as well as dis­persion and dissipation errors as a function of wavenumber. Afterwards, I experimentally investigate their relative suitability towards scale­-resolving simulation of compressible and turbulent flows, by solv­ing a number of simple test cases of increasing difficulty (linear advection, inviscid Burgers and Euler equations; all in 1D) using a purpose­-made MATLAB implementation. The proposed isogeometric method (DGIGA) has been found to be at least as viable as the other two, a priori, for the resolution of high­-speed turbulent flows. Moreover, I have found that low dispersion and dissipation need not always be associated to high order, but to a high number of degrees of freedom per patch instead; these two coincide in the more conventional schemes, yet not necessarily in DGIGA. As a nonlinear stabilization mechanism, however, the proposed combination of DGIGA with AFC has turned out to be inferior to existing limiters.