The instationary Stokes and Navier−Stokes equations are considered in a simultaneously space-time variational saddle point formulation, so involving both velocities u and pressure p. For the instationary Stokes problem, it is shown that the corresponding operator is a boundedly invertible linear mapping between H1 and H'2, both Hilbert spaces H1 and H2 being Cartesian products of (intersections of) Bochner spaces, or duals of those. Based on these results, the operator that corresponds to the Navier−Stokes equations is shown to map H1 into H'2, with a Fréchet derivative that, at any (u,p) ∈ H1, is boundedly invertible. These results are essential for the numerical solution of the combined pair of velocities and pressure as function of simultaneously space and time. Such a numerical approach allows for the application of (adaptive) approximation from tensor products of spatial and temporal trial spaces, with which the instationary problem can be solved at a computational complexity that is of the order as for a corresponding stationary problem.

We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a δ-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on δ. We apply the obtained estimates to show exponential convergence with rate O(exp(-b√N)), N being the number of degrees of freedom and b > 0, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate O(exp(-b3√N)), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.

We analyze the regularity of random entropy solutions to scalar hyperbolic conservation laws with random initial data. We prove regularity theorems for statistics of random entropy solutions like expectation, variance, space-time correlation functions and polynomial moments such as gPC coefficients. We show how regularity of such moments (statistical and polynomial chaos) of random entropy solutions depends on the regularity of the distribution law of the random shock location of the initial data. Sufficient conditions on the law of the initial data for moments of the random entropy solution to be piece-wise smooth functions of space and time are identified, even in cases where path-wise random entropy solutions are discontinuous almost surely. We extrapolate the results to hyperbolic systems of conservation laws in one space dimension. We then exhibit a class of stochastic Galerkin discretizations which allows to derive closed deterministic systems of hyperbolic conservation laws for the coefficients in truncated polynomial chaos expansions of the random entropy solution. Based on the regularity theory developed here, we show that depending on the smoothness of the law of the initial data, arbitrarily high convergence rates are possible for the computation of coefficients in gPC approximations of random entropy solutions for Riemann problems with random shock location by combined Stochastic Galerkin Finite Volume schemes.

The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated in the Hilbert space V = H01(D) by multivariate sparse polynomials in the parameter vector y with a controlled number N of terms. The convergence rate in terms of N does not depend on the number of parameters in V, which may be arbitrarily large or countably infinite, thereby breaking the curse of dimensionality. However, these approximation results do not describe the concrete construction of these polynomial expansions, and should therefore rather be viewed as benchmark for the convergence analysis of numerical methods. The present paper presents an adaptive numerical algorithm for constructing a sequence of sparse polynomials that is proved to converge toward the solution with the optimal benchmark rate. Numerical experiments are presented in large parameter dimension, which confirm the effectiveness of the adaptive approach.

The ‘holy grail’ of exoplanet research today is the detection of an earth-like planet: a rocky planet in the habitable zone around a main-sequence star. Extremely precise Doppler spectroscopy is an indispensable tool to find and characterize earth-like planets; however, to find these planets around solar-type stars, we need nearly one order of magnitude better radial velocity (RV) precision than the best current spectrographs provide. Recent developments in astrophotonics (Bland-Hawthorn & Horton 2006, Bland-Hawthorn et al. 2010) and adaptive optics (AO) enable single mode fiber (SMF) fed, high resolution spectrographs, which can realize the next step in precision. SMF feeds have intrinsic advantages over multimode fiber or slit coupled spectrographs: The intensity distribution at the fiber exit is extremely stable, and as a result the line spread function of a well-designed spectrograph is fully decoupled from input coupling conditions, like guiding or seeing variations (Ihle et al. 2010). Modal noise, a limiting factor in current multimode fiber fed instruments (Baudrand & Walker 2001), can be eliminated by proper design, and the diffraction limited input to the spectrograph allows for very compact instrument designs, which provide excellent optomechanical stability. A SMF is the ideal interface for new, very precise wavelength calibrators, like laser frequency combs (Steinmetz et al. 2008, Osterman et al. 2012), or SMF based Fabry-Perot Etalons (Halverson et al. 2013). At near infrared wavelengths, these technologies are ready to be implemented in on-sky instruments, or already in use. We discuss a novel concept for such a spectrograph.

Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of best N-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficients.

Galerkin discretizations of integral equations in $\mathbb{R}^{d}$ require the evaluation of integrals $I = \int_{S^{(1)}}\int_{S^{(2)}}g(x,y){\rm d}y{\rm d}x$ where S(1),S(2) are d-simplices and g has a singularity at x = y. We assume that g is Gevrey smooth for x$\ne$y and satisfies bounds for the derivatives which allow algebraic singularities at x = y. This holds for kernel functions commonly occurring in integral equations. We construct a family of quadrature rules $\mathcal{Q}_{N}$ using N function evaluations of g which achieves exponential convergence |I – $\mathcal{Q}_{N}$| ≤C exp(–rNγ) with constants r, γ > 0.

We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form $-a:\nabla\nabla u + b \cdot \nabla u + cu = f(x)$, $x \in \Omega = (0,1)^d \subset \mathbb{R}^d$, where $a \in \mathbb{R}^{d\times d}$ is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse finite element approximation uh on a partition of Ω of mesh size h = hL = 2-L satisfies the following bound in the streamline-diffusion norm $|||\cdot|||_{\rm SD}$, provided u belongs to the space $\mathcal{H}^{k+1}(\Omega)$ of functions with square-integrable mixed (k+1)st derivatives: \[ |||u-u_h|||_{\rm SD}\leq C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\} (|\sqrt{a}| h_L^t + |b|^{\frac{1}{2}} h_L^{t+\frac{1}{2}} + c^{\frac{1}{2}} h_L^{t+1} \!)|u|_{\mathcal{H}^{t+1}(\Omega)}, \qquad \qquad \qquad \] where $\kappa_i=\kappa_i(p,t,L)$, i=0,1, and $1 \leq t \leq \min(k,p)$. We show, under various mild conditions relating L to p, L to d, or p to d, that in the case of elliptic transport-dominated diffusion problems $\kappa_0, \kappa_1 \in (0,1)$, and hence for p ≥ 1 the 'error constant' $C_{p,t} d^2 \max\{(2-p)_+,\kappa_0^{d-1},\kappa_1^d\}$ exhibits exponential decay as d → ∞; in the case of a general symmetric positive semidefinite matrix a, the error constant is shown to grow no faster than $\mathcal{O}(d^2)$. In any case, in the absence of assumptions that relate L, p and d, the error $|||u - u_h|||_{\rm SD}$ is still bounded by $\kappa_\ast^{d-1} |\log_2 h_L|^{d-1}\mathcal{O}(|\sqrt{a}| h_L^t + |b|^{\frac{1}{2}} h_L^{t+\frac{1}{2}} + c^{\frac{1}{2}} h_L^{t+1})$, where $\kappa_\ast \in (0,1)$ for all L, p, d ≥ 2.

Organic light-emitting diodes (OLED) offer the potential to replace conventional light sources such as incandescent bulbs and fluorescent tubes. The question which thin-film technology is most favorable to produce OLED on an industrial scale is still unanswered. The most established technology for the deposition of small-molecule organic layers is vacuum thermal evaporation. A comparably novel technology is organic vapor phase deposition (OVPD), which offers some unique features in terms of adjustable process parameters such as deposition chamber pressure (P) and substrate temperature (TS). The impact of these parameters on the morphology of organic single layers as well as on the performance of OLED is mostly unknown. In this work, phosphorescent red OLED were produced with different TS and a strong influence on the device efficiency was found. Atomic force microscopy measurements were conducted to investigate the morphology of the hole injection and hole transport layers of the devices deposited at different TS. In addition to this, the influence of TS and P on the performance of fluorescent blue OLED and the morphology of organic single layers was tested. By varying TS and P for the emission layer only, current efficiencies in the range from 4.3 to 6.8 cd/A were found despite the fact that all devices had the same structure. Atomic force microscopy measurements conducted on organic single layers which were deposited at the same process conditions showed rms values ranging from 1.4 to 57 nm.

Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) $\partial_t u + {\mathcal{A}}[u] = 0$. This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for ${\mathcal{A}}$ can be replaced by a sparse matrix in the wavelet basis, and the linear systems in each implicit time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O(MN(log(N))2) operations and O(Nlog(N)) memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black–Scholes equation. Computational examples for various Lévy price processes are presented.

We consider the numerical solution of diffusion problems in (0,T) x Ω for $\Omega\subset \mathbb{R}^d$ and for T > 0 in dimension dd ≥ 1. We use a wavelet based sparse grid space discretization with mesh-width h and order pd ≥ 1, and hp discontinuous Galerkin time-discretization of order $r = O(\left|\log h\right|)$ on a geometric sequence of $O(\left|\log h\right|)$ many time steps. The linear systems in each time step are solved iteratively by $O(\left|\log h\right|)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L2(Ω)-error of O(N-p) for u(x,T) where N is the total number of operations, provided that the initial data satisfies $u_0 \in H^\varepsilon(\Omega)$ with ε > 0 and that u(x,t) is smooth in x for t>0. Numerical experiments in dimension d up to 25 confirm the theory.

In this work, we analyze hierarchic hp-finite element discretizations of the full, three-dimensional plate problem. Based on two-scale asymptotic expansion of the three-dimensional solution, we give specific mesh design principles for the hp-FEM which allow to resolve the three-dimensional boundary layer profiles at robust, exponential rate. We prove that, as the plate half-thickness ε tends to zero, the hp-discretization is consistent with the three-dimensional solution to any power of ε in the energy norm for the degree $p={\cal O}(\left|{\log \varepsilon}\right|)$ and with ${\cal O}({p^4})$ degrees of freedom.

The convergence of a two-scale FEM for elliptic problems in divergence form with coefficients and geometries oscillating at length scale ε << 1 is analyzed. Full elliptic regularity independent of ε is shown when the solution is viewed as mapping from the slow into the fast scale. Two-scale FE spaces which are able to resolve the ε scale of the solution with work independent of ε and without analytical homogenization are introduced. Robust in ε error estimates for the two-scale FE spaces are proved. Numerical experiments confirm the theoretical analysis.