A posteriori error estimates for leap-frog and cosine methods for second order evolution problems

We consider second order explicit and implicit two-step time-discrete schemes for wave-type equations. We derive optimal order aposteriori estimates controlling the time discretization error. Our analysis, has been motivated by the need to provide aposteriori estimates for the popular leap-frog method (also known as Verlet's method in molecular dynamics literature); it is extended, however, to general cosine-type second order methods. The estimators are based on a novel reconstruction of the time-dependent component of the approximation. Numerical experiments confirm similarity of convergence rates of the proposed estimators and of the theoretical convergence rate of the true error

Space-time Methods for Time-dependent Partial Differential Equations

Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space. Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations

Asymptotically constant-free and polynomial-degree-robust a posteriori estimates for space discretizations of the wave equation

We derive an equilibrated a posteriori error estimator for the space (semi) discretization of the scalar wave equation by finite elements. In the idealized setting where time discretization is ignored and the simulation time is large, we provide fully-guaranteed upper bounds that are asymptotically constant-free and show that the proposed estimator is efficient and polynomial-degree-robust, meaning that the efficiency constant does not deteriorate as the approximation order is increased. To the best of our knowledge, this work is the first to derive provably efficient error estimates for the wave equation. We also explain, without analysis, how the estimator is adapted to cover time discretization by an explicit time integration scheme. Numerical examples illustrate the theory and suggest that it is sharp

A Comparison of Two Shallow Water Models with Non-Conforming Adaptive Grids: classical tests

In an effort to study the applicability of adaptive mesh refinement (AMR) techniques to atmospheric models an interpolation-based spectral element shallow water model on a cubed-sphere grid is compared to a block-structured finite volume method in latitude-longitude geometry. Both models utilize a non-conforming adaptation approach which doubles the resolution at fine-coarse mesh interfaces. The underlying AMR libraries are quad-tree based and ensure that neighboring regions can only differ by one refinement level. The models are compared via selected test cases from a standard test suite for the shallow water equations. They include the advection of a cosine bell, a steady-state geostrophic flow, a flow over an idealized mountain and a Rossby-Haurwitz wave. Both static and dynamics adaptations are evaluated which reveal the strengths and weaknesses of the AMR techniques. Overall, the AMR simulations show that both models successfully place static and dynamic adaptations in local regions without requiring a fine grid in the global domain. The adaptive grids reliably track features of interests without visible distortions or noise at mesh interfaces. Simple threshold adaptation criteria for the geopotential height and the relative vorticity are assessed.Comment: 25 pages, 11 figures, preprin

An adaptive finite element method for high-frequency scattering problems with variable coefficients

We introduce a new method for the numerical approximation of time-harmonic acoustic scattering problems stemming from material inhomogeneities. The method works for any frequency $\omega$, but is especially efficient for high-frequency problems. It is based on a time-domain approach and consists of three steps: \emph{i)} computation of a suitable incoming plane wavelet with compact support in the propagation direction; \emph{ii)} solving a scattering problem in the time domain for the incoming plane wavelet; \emph{iii)} reconstruction of the time-harmonic solution from the time-domain solution via a Fourier transform in time. An essential ingredient of the new method is a front-tracking mesh adaptation algorithm for solving the problem in \emph{ii)}. By exploiting the limited support of the wave front, this allows us to make the number of the required degrees of freedom to reach a given accuracy significantly less dependent on the frequency $\omega$, as shown in the numerical experiments. We also present a new algorithm for computing the Fourier transform in \emph{iii)} that exploits the reduced number of degrees of freedom corresponding to the adapted meshes

Potential fields data modeling: new frontiers in forward and inverse problems

Since the '50, potential fields data modeling has played an important role in analyzing the density and magnetization distribution in Earth's subsurface for a wide variety of applications. Examples are the characterization of ore deposits and the assessment of geothermal and petroleum potential, which turned out to be key contributors for the economic and industrial development after World War II. Current modeling methods mainly rely on two popular parameterization approaches, either involving a discretization of target geological bodies by means of 2D to 2.75D horizontal prisms with polygonal vertical cross-section (polygon-based approach) or prismatic cells (prism-based approach). Despite the great endeavour made by scientists in recent decades, inversion methods based on these parameterization approaches still suffers from a limited ability to (i) realistically characterize the variability of density and magnetization expected in a study area and (ii) take into account the strong non-uniqueness affecting potential fields theory. The prism-based approach is used in linear deterministic inverse methods, which provide just one single solution, preventing uncertainty estimation and statistical analysis on the parameters we would like to characterize (i.e, density or magnetization). On the contrary, the polygon-based approach is almost exclusively exploited in a trial-and-error modeling strategy, leaving the potential to develop innovative inverse methods untapped. The reason is two-fold, namely (i) its strongly non-linear forward problem requires an efficient probabilistic inverse modeling methodology to solve the related inverse problem, and (ii) unpredictable cross-intersections between polygonal bodies during inversion represent a challenging task to be tackled in order to achieve geologically plausible model solutions. The goal of this thesis is then to contribute to solving the critical issues outlined above, developing probabilistic inversion methodologies based on the polygon- and prism-based parameterization approaches aiming to help improving our capability to unravel the structure of the subsurface. Regarding the polygon-based parameterization strategy, at first a deep review of its mathematical framework has been performed, allowing us (i) to restore the validity of a recently criticized mathematical formulation for the 2D magnetic case, and (ii) to find an error sign in the derivation for the 2.75D magnetic case causing potentially wrong numerical results. Such preliminary phase allowed us to develop a methodology to independently or jointly invert gravity and magnetic data exploiting the Hamilton Monte Carlo approach, thanks to which collection of models allow researchers to appraise different geological scenarios and fully characterize uncertainties on the model parameters. Geological plausibility of results is ensured by automatic checks on the geometries of modelled bodies, which avoid unrealistic cross-intersections among them. Regarding the prism-based parameterization approach, the linear inversion method based on the probabilistic approach considers a discretization of target geological scenarios by prismatic bodies, arranged horizontally to cover it and finitely extended in the vertical direction, particularly suitable to model density and magnetization variability inside strata. Its strengths have been proven, for the magnetic case, in the characterization of the magnetization variability expected for the shallower volcanic unit of the Mt. Melbourne Volcanic Field (Northern Victoria Land, Antarctica), helping significantly us to unravel its poorly known inner geophysical architecture