Space-time finite element methods for parabolic problems

Abstract We propose and analyze a space-time finite element method for the numerical solution of parabolic evolution equations. This approach allows the use of general and unstructured space-time finite elements which do not require any tensor product structure. The stability of the numerical scheme is based on a stability condition which holds for standard finite element spaces. We also provide related a priori error estimates which are confirmed by numerical experiments.</jats:p

Space-Time Discretizations Using Constrained First-Order System Least Squares (CFOSLS)

This paper studies finite element discretizations for three types of time-dependent PDEs, namely heat equation, scalar conservation law and wave equation, which we reformulate as first order systems in a least-squares setting subject to a space-time conservation constraint (coming from the original PDE). Available piece- wise polynomial finite element spaces in (n + 1)-dimensions for functional spaces from the (n + 1)-dimensional de Rham sequence for n = 3, 4 are used for the implementation of the method. Computational results illustrating the error behavior, iteration counts and performance of block-diagonal and monolithic geometric multi- grid preconditioners are presented for the discrete CFOSLS system. The results are obtained from a parallel implementation of the methods for which we report reasonable scalability

Asynchronous parallel solver for hyperbolic problems via the Spacetime Discontinuous Galerkin method

This thesis presents a parallel Space Time Discontinuous Galerkin (SDG) finite element method which makes use of the method's unstructured mesh generation and localized solution technique to achieve a high level of parallel scalability. Our SDG method is different from most traditional adaptive finite element methods in that the solution process generates fully unstructured spacetime grids that satisfy a special causality constraint ensuring that computations can occur locally on small cluster of spacetime elements. The resulting asynchronous solution scheme offers several desirable features: element-wise conservation of solution quantities, strong stability properties without the need for explicit stabilization, local mesh adaptivity operations and linear complexity in the number of spacetime elements. In this thesis we propose an algorithm that effectively parallelizes the Tent Pitcher algorithm developed by [1] using the POSIX Thread (or Pthread) parallel execution model. Multiple software threads can simultaneously and asynchronously perform patch computations by advancing vertices in time. By enforcing the causality constraint on the time step, we can guarantee that each thread only performs calculations using data computed previously. Additionally, improvements to the adaptivity scheme allow for local mesh refinement and coarsening while maintaining globally conforming triangulation. Numerical tests show that our algorithm achieves high parallel scalability using shared-memory parallelization

Unstructured Grid Adaptation: Status, Potential Impacts, and Recommended Investments Towards CFD 2030

International audienceUnstructured grid adaptation is a powerful tool to control Computational Fluid Dynamics (CFD) discretization error. It has enabled key increases in the accuracy, automation, and capacity of some fluid simulation applications. Slotnick et al. provide a number of case studies in the CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences to illustrate the current state of CFD capability and capacity. The study authors forecast the potential impact of emerging High Performance Computing (HPC) environments forecast in the year 2030 and identify that mesh generation and adaptivity will continue to be significant bottlenecks in the CFD workflow. These bottlenecks may persist because very little government investment has been targeted in these areas. To motivate investment, the impacts of improved grid adaptation technologies are identified. The CFD Vision 2030 Study roadmap and anticipated capabilities in complementary disciplines are quoted to provide context for the progress made in grid adaptation in the past fifteen years, current status, and a forecast for the next fifteen years with recommended investments. These investments are specific to mesh adaptation and impact other aspects of the CFD process. Finally, a strategy is identified to di↵use grid adaptation technology into production CFD work flows

Selected topics in Computational Relativity

This thesis addresses a collection of topics that are either directly related to, or have implications for, current challenges in computational relativity. In the first part, we explore a spacetime discretization method for computational relativity. This offers unique computational advantages, for distributing the computation over a large number of processes, as well as for studying spacetime regions close to black hole singularities. In the second part, we present a method to construct initial conditions for numerical evolution of charged, spinning black hole binaries. The evolution of these initial conditions provides a proxy for binary black hole waveforms in modified theories of gravity. In the third part of the thesis, we focus on building an empirical understanding of why Boolean Satisfiability (SAT) solvers are efficient for real-world problems, when, theoretically, the Boolean SAT problem is computationally intractable

Adaptive spacetime meshing for discontinuous Galerkin methods

Spacetime-discontinuous Galerkin (SDG) finite element methods are used to solve hyperbolic spacetime partial differential equations (PDEs) to accurately model wave propagation phenomena arising in important applications in science and engineering. Tent Pitcher is a specialized algorithm, invented by Üngör and Sheffer (2000) and extended by Erickson et al. (2005) to construct an unstructured simplicial (d + 1)-dimensional spacetime mesh over an arbitrary d-dimensional space domain. Tent Pitcher is an advancing front algorithm that incrementally adds groups of elements to the evolving spacetime mesh. It supports an accurate, local, and parallelizable solution strategy by interleaving mesh generation with an SDG solver. When solving nonlinear PDEs, previous versions of Tent Pitcher must make conservative worst-case assumptions about the physical parameters which limit the duration of spacetime elements. Thus, these algorithms create a mesh with many more elements than necessary. In this paper, we extend Tent Pitcher to give the first spacetime meshing algorithm suitable for efficient simulation of nonlinear phenomena using SDG methods. We adapt the duration of spacetime elements to changing physical parameters due to nonlinear response. Given a triangulated 2-dimensional Euclidean space domain M corresponding to time t = 0 and initial and boundary conditions of the underlying hyperbolic spacetime PDE, we construct an unstructured tetrahedral mesh in the spacetime domain E^2 x R. For every target time T ≥ 0, our algorithm meshes the spacetime volume M x [0, T] with a bounded number of non-degenerate tetrahedra. A recent extension of Tent Pitcher due to Abedi et al. (2004) adapts the spatial size of spacetime elements in 2D x time to a posteriori estimates of numerical error. Our extension of Tent Pitcher retains the ability to perform adaptive refinement and coarsening of the mesh. We thus obtain the first adaptive nonlinear Tent Pitcher algorithm to build spacetime meshes in 2D x time

Discrétisation Espace-Temps d'Équations d'Ondes Élasto-Acoustiques dans des Bases Trefftz-DG Polynomiales

Discontinuous Finite Element Methods (DG FEM) have proven flexibility and accuracy for solving wave problems in complex media. However, they require a large number of degrees of freedom, which increases the corresponding computational cost compared with that of continuous finite element methods. Among the different variational approaches to solve boundary value problems, there exists a particular family of methods, based on the use of trial functions in the form of exact local solutions of the governing equations. The idea was first proposed by Trefftz in 1926, and since then it has been further developed and generalized. A Trefftz-DG variational formulation applied to wave problems reduces to surface integrals that should contribute to decreasing the computational costs.Trefftz-type approaches have been widely used for time-harmonic problems, while their implementation for time-dependent simulations is still limited. The feature of Trefftz-DG methods applied to time-dependent problems is in the use of space-time meshes. Indeed, standard DG methods lead to the construction of a semi-discrete system of ordinary differential equations in time which are integrated by using an appropriate scheme. But Trefftz-DG methods applied to wave problems lead to a global matrix including time and space discretizations which is huge and sparse. This significantly hampers the deployment of this technology for solving industrial problems.In this work, we develop a Trefftz-DG framework for solving mechanical wave problems including elasto-acoustic equations. We prove that the corresponding formulations are well-posed and we address the issue of solving the global matrix by constructing an approximate inverse obtained from the decomposition of the global matrix into a block-diagonal one. The inversion is then justified under a CFL-type condition. This idea allows for reducing the computational costs but its accuracy is limited to small computational domains. According to the limitations of the method, we have investigated the potential of Tent Pitcher algorithms following the recent works of Gopalakrishnan et al. It consists in constructing a space-time mesh made of patches that can be solved independently under a causality constraint. We have obtained very promising numerical results illustrating the potential of Tent Pitcher in particular when coupled with a Trefftz-DG method involving only surface terms. In this way, the space-time mesh is composed of elements which are 3D objects at most. It is also worth noting that this framework naturally allows for local time-stepping which is a plus to increase the accuracy while decreasing the computational burden.Les méthodes d'éléments finis de type Galerkine discontinu (DG FEM) ont démontré précision et efficacité pour résoudre des problèmes d'ondes dans des milieux complexes. Cependant, elles nécessitent un très grand nombre de degrés de liberté, ce qui augmente leur coût de calcul en comparaison du coût des méthodes d'éléments finis continus. Parmi les différentes approches variationnelles pour résoudre les problèmes aux limites, se distingue une famille particulière, basée sur l'utilisation de fonctions tests qui sont des solutions locales exactes des équations à résoudre. L'idée vient de E.Trefftz en 1926 et a depuis été largement développée et généralisée. Les méthodes variationnelles de type Trefftz-DG appliquées aux problèmes d'ondes se réduisent à des intégrales de surface, ce qui devrait contribuer à réduire les coûts de calcul.Les approches de type Trefftz ont été largement développées pour les problèmes harmoniques, mais leur utilisation pour des simulations en domaine transitoire est encore limitée. Quand elles sont appliquées dans le domaine temporel, les méthodes de Trefftz utilisent des maillages qui recouvrent le domaine espace-temps. C'est une des paraticularités de ces méthodes. En effet, les méthodes DG standards conduisent à la construction d'un système semi-discret d'équations différentielles ordinaires en temps qu'on intègre avec un schéma en temps explicite. Mais les méthodes de Trefftz-DG appliquées aux problèmes d'ondes conduisent à résoudre une matrice globale, contenant la discrétisation en espace et en temps, qui est de grande taille et creuse. Cette particularité gêne considérablement le déploiement de cette technologie pour résoudre des problèmes industriels.Dans ce travail, nous développons un environnement Trefftz-DG pour résoudre des problèmes d'ondes mécaniques, y compris les équations couplées de l'élasto-acoustique. Nous prouvons que les formulations obtenues sont bien posées et nous considérons la difficulté d'inverser la matrice globale en construisant un inverse approché obtenu à partir de la décomposition de la matrice globale en une matrice diagonale par blocs. Cette idée permet de réduire les coûts de calcul mais sa précision est limitée à de petits domaines de calcul. Etant données les limitations de la méthode, nous nous sommes intéressés au potentiel du "Tent Pitcher", en suivant les travaux récents de Gopalakrishnan et al. Il s'agit de construire un maillage espace-temps composé de macro-éléments qui peuvent être traités indépendamment en faisant une hypothèse de causalité. Nous avons obtenu des résultats préliminaires très encourageants qui illustrent bien l'intérêt du Tent Pitcher, en particulier quand il est couplé à une méthode de Trefftz-DG formulée à partir d'intégrales de surface seulement. Dans ce cas, le maillage espace-temps est composé d'éléments qui sont au plus de dimension 3. Il est aussi important de noter que ce cadre se prête à l'utilisation de pas de temps locaux ce qui est un plus pour gagner en précision avec des coûts de calcul réduits