The DUNE-ALUGrid Module

In this paper we present the new DUNE-ALUGrid module. This module contains a major overhaul of the sources from the ALUgrid library and the binding to the DUNE software framework. The main changes include user defined load balancing, parallel grid construction, and an redesign of the 2d grid which can now also be used for parallel computations. In addition many improvements have been introduced into the code to increase the parallel efficiency and to decrease the memory footprint. The original ALUGrid library is widely used within the DUNE community due to its good parallel performance for problems requiring local adaptivity and dynamic load balancing. Therefore, this new model will benefit a number of DUNE users. In addition we have added features to increase the range of problems for which the grid manager can be used, for example, introducing a 3d tetrahedral grid using a parallel newest vertex bisection algorithm for conforming grid refinement. In this paper we will discuss the new features, extensions to the DUNE interface, and explain for various examples how the code is used in parallel environments.Comment: 25 pages, 11 figure

Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status solidi (b) - basic solid state physics" devoted to the CECAM workshop "State of the art developments and perspectives of real-space electronic structure techniques in condensed matter and molecular physics". v2: Minor stylistic and typographical changes, partly inspired by referee comment

High-order composite finite element exact sequences based on tetrahedral-hexahedral-prismatic-pyramidal partitions

The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces. This paper presents high order exact sequences of finite element approximations in H^1 (Ω), H(curl, Ω), H(div, Ω), and L^2(Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy's problems based on compatible pairs of approximations in {H(div, Ω), L^2 (Ω)} for such tetrahedral-hexahedral-prismatic-pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables are obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems

Construction of higher-order curl-conforming finite elements and its assembly

Different choices are available when constructing vector finite element bases in real coordinates. In this communication, two different designs of higher-order curl-conforming basis functions are introduced and explained, showing the particularities of its assembly. Tetrahedra and hexahedra are used as element shapes to assess the effect of triangular and quadrilateral faces on the two considered constructions of basis functions. A comparison of their robustness in terms of the condition number of the finite element matrices for a number of distortions is includedMinisterio de Ciencia y Tecnología, Grant/Award Numbers: TEC2013- 47753-C3, TEC2016-80386-P; Ministerio de Educación, Cultura y Deporte, Grant/Award Number: FPU14/0374

Doctor of Philosophy

dissertationPartial differential equations (PDEs) are widely used in science and engineering to model phenomena such as sound, heat, and electrostatics. In many practical science and engineering applications, the solutions of PDEs require the tessellation of computational domains into unstructured meshes and entail computationally expensive and time-consuming processes. Therefore, efficient and fast PDE solving techniques on unstructured meshes are important in these applications. Relative to CPUs, the faster growth curves in the speed and greater power efficiency of the SIMD streaming processors, such as GPUs, have gained them an increasingly important role in the high-performance computing area. Combining suitable parallel algorithms and these streaming processors, we can develop very efficient numerical solvers of PDEs. The contributions of this dissertation are twofold: proposal of two general strategies to design efficient PDE solvers on GPUs and the specific applications of these strategies to solve different types of PDEs. Specifically, this dissertation consists of four parts. First, we describe the general strategies, the domain decomposition strategy and the hybrid gathering strategy. Next, we introduce a parallel algorithm for solving the eikonal equation on fully unstructured meshes efficiently. Third, we present the algorithms and data structures necessary to move the entire FEM pipeline to the GPU. Fourth, we propose a parallel algorithm for solving the levelset equation on fully unstructured 2D or 3D meshes or manifolds. This algorithm combines a narrowband scheme with domain decomposition for efficient levelset equation solving

New higher-order basis functions for Curvilinear finite elements

The first contribution is a fast calculation method for tetrahedral finite element matrices which is applicable to curvilinear geometries and inhomogeneous material properties. The element matrices are obtained at a low computational cost via scaled additions of universal matrices. The proposed technique is more efficient than competing approaches and provides well-defined lower and upper bounds for the required number of matrices. In the case of tetrahedral H(div) elements, a new set of basis functions is proposed for the mixed-order Nédélec space. The specialty of the functions is a high level of orthogonality which applies to arbitrary straight-sided tetrahedra. The resulting condition numbers, compared to competing bases, are significantly lower. The remaining contributions concern hexahedral elements, where a new, mixed-order serendipity element is proposed for H(curl)-conforming functions. It allows the construction of a single set of hierarchical basis functions that can also be used to span various other finite element spaces. Therefore, it is possible to use different finite element spaces within the same mesh while maintaining conformity. In the curvilinear case, a special yet versatile way of mesh refinement is proposed along with serendipity basis functions for the interpolation of the geometry. The main advantage of the proposed methods is the resulting algebraic rate of convergence in H(curl)-norm with the least possible number of unknowns.Der erste Beitrag ist eine schnelle Berechnungsmethode von Finite-Elemente-Matrizen für Tetraeder, die auf krummlinige Geometrien und inhomogene Materialeigenschaften anwendbar ist. Die Elementmatrizen werden mit geringem Rechenaufwand durch skalierte Addition vorgefertigter Matrizen erstellt. Die vorgeschlagene Methode ist effizienter als vergleichbare Ansätze und liefert wohldefinierte obere und untere Schranken für die Anzahl der benötigten Matrizen. Für H(div)-konforme Elemente auf Tetraedern werden neue Ansatzfunktionen für den N´ed´elec-Raum gemischter Ordnung vorgestellt. Die Besonderheit dieser Funktionen ist ein hohes Maß an Orthogonalität für beliebige geradlinige Tetraeder. Im Vergleich zu anderen Ansatzfunktionen sind die resultierenden Konditionszahlen deutlich kleiner. Die übrigen Beiträge betreffen Hexaeder, für die ein neues Serentipity-Element gemischter Ordnung vorgestellt wird. Es ermöglicht die Konstruktion hierarchischer Ansatzfunktionen, die auch zum Aufspannen anderer Finite-Elemente-Räume angewandt werden kann. Daher ist es möglich, verschiedene Finite-Elemente-Räume auf dem gleichen Netz zu verwenden und dabei Konformität zu bewahren. Für den krummlinigen Fall wird eine spezielle aber vielseitige Methode zur Netzverfeinerung mit Serentipity-Ansatzfunktionen zur Interpolation der Geometrie vorgestellt. Der Hauptvorteil der vorgestellten Methoden ist die algebraische Konvergenz in der Norm des H(rot) mit der kleinstmöglichen Anzahl an Unbekannten