355 research outputs found

    High-Order Mixed Finite Element Variable Eddington Factor Methods

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    We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the Discrete Ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory. Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations

    Fast boundary element methods for the simulation of wave phenomena

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    This thesis is concerned with the efficient implementation of boundary element methods (BEM) for their application in wave problems. BEM present a particularly useful tool, since they reduce the dimension of the problems by one, resulting in much fewer unknowns. However, this comes at the cost of dense system matrices, whose entries require the integration of singular kernel functions over pairs of boundary elements. Because calculating these four-dimensional integrals by cubature rules is expensive, a novel approach based on singularity cancellation and analytical integration is proposed. In this way, the dimension of the integrals is reduced and closed formulae are obtained for the most challenging cases. This allows for the accurate calculation of the matrix entries while requiring less computational work compared with conventional numerical integration. Furthermore, a new algorithm based on hierarchical low-rank approximation is presented, which compresses the dense matrices and improves the complexity of the method. The idea is to collect the matrices corresponding to different time steps in a third-order tensor and to approximate individual sub-blocks by a combination of analytic and algebraic low-rank techniques. By exploiting the low-rank structure in several ways, the method scales almost linearly in the number of spatial degrees of freedom and number of time steps. The superior performance of the new method is demonstrated in numerical examples.Diese Arbeit befasst sich mit der effizienten Implementierung von Randelementmethoden (REM) fĂŒr ihre Anwendung auf Wellenprobleme. REM stellen ein besonders nĂŒtzliches Werkzeug dar, da sie die Dimension der Probleme um eins reduzieren, was zu weit weniger Unbekannten fĂŒhrt. Allerdings ist dies mit vollbesetzten Matrizen verbunden, deren EintrĂ€ge die Integration singulĂ€rer Kernfunktionen ĂŒber Paare von Randelementen erfordern. Da die Berechnung dieser vierdimensionalen Integrale durch Kubaturformeln aufwendig ist, wird ein neuer Ansatz basierend auf Regularisierung und analytischer Integration verfolgt. Auf diese Weise reduziert sich die Dimension der Integrale und es ergeben sich geschlossene Formeln fĂŒr die schwierigsten FĂ€lle. Dies ermöglicht die genaue Berechnung der MatrixeintrĂ€ge mit geringerem Rechenaufwand als konventionelle numerische Integration. Außerdem wird ein neuer Algorithmus beruhend auf hierarchischer Niedrigrangapproximation prĂ€sentiert, der die Matrizen komprimiert und die KomplexitĂ€t der Methode verbessert. Die Idee ist, die Matrizen der verschiedenen Zeitpunkte in einem Tensor dritter Ordnung zu sammeln und einzelne Teilblöcke durch eine Kombination von analytischen und algebraischen Niedrigrangverfahren zu approximieren. Durch Ausnutzung der Niedrigrangstruktur skaliert die Methode fast linear mit der Anzahl der rĂ€umlichen Freiheitsgrade und der Anzahl der Zeitschritte. Die ĂŒberlegene Leistung der neuen Methode wird anhand numerischer Beispiele aufgezeigt

    New lower order mixed finite element methods for linear elasticity

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    New lower order H(div)H(\textrm{div})-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the (d+1)(d+1)-order normal-normal face bubble space. The reduced counterpart has only d(d+1)2d(d+1)^2 degrees of freedom. In two dimensions, basis functions are explicitly given in terms of barycentric coordinates. Lower order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lam\'{e} coefficient λ\lambda, and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates.Comment: 23 pages, 2 figure

    Exploiting spatial symmetries for solving Poisson's equation

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    This paper presents a strategy to accelerate virtually any Poisson solver by taking advantage of s spatial reflection symmetries. More precisely, we have proved the existence of an inexpensive block diagonalisation that transforms the original Poisson equation into a set of 2s fully decoupled subsystems then solved concurrently. This block diagonalisation is identical regardless of the mesh connectivity (structured or unstructured) and the geometric complexity of the problem, therefore applying to a wide range of academic and industrial configurations. In fact, it simplifies the task of discretising complex geometries since it only requires meshing a portion of the domain that is then mirrored implicitly by the symmetries’ hyperplanes. Thus, the resulting meshes naturally inherit the exploited symmetries, and their memory footprint becomes 2s times smaller. Thanks to the subsystems’ better spectral properties, iterative solvers converge significantly faster. Additionally, imposing an adequate grid points’ ordering allows reducing the operators’ footprint and replacing the standard sparse matrix-vector products with the sparse matrixmatrix product, a higher arithmetic intensity kernel. As a result, matrix multiplications are accelerated, and massive simulations become more affordable. Finally, we include numerical experiments based on a turbulent flow simulation and making state-of-theart solvers exploit a varying number of symmetries. On the one hand, algebraic multigrid and preconditioned Krylov subspace methods require up to 23% and 72% fewer iterations, resulting in up to 1.7x and 5.6x overall speedups, respectively. On the other, sparse direct solvers’ memory footprint, setup and solution costs are reduced by up to 48%, 58% and 46%, respectively.This work has been financially supported by two competitive R+D projects: RETOtwin (PDC2021-120970-I00), given by MCIN/AEI/10.13039/501100011033 and European Union Next GenerationEU/PRTR, and FusionCAT (001-P-001722), given by Generalitat de Catalunya RIS3CAT-FEDER. Àdel Alsalti-Baldellou has also been supported by the predoctoral grants DIN2018-010061 and 2019-DI-90, given by MCIN/AEI/10.13039/501100011033 and the Catalan Agency for Management of University and Research Grants (AGAUR), respectively.Peer ReviewedPostprint (published version

    Pressure-robust enriched Galerkin methods for the Stokes equations

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    In this paper, we present a pressure-robust enriched Galerkin (EG) scheme for solving the Stokes equations, which is an enhanced version of the EG scheme for the Stokes problem proposed in [Son-Young Yi, Xiaozhe Hu, Sanghyun Lee, James H. Adler, An enriched Galerkin method for the Stokes equations, Computers and Mathematics with Applications, accepted, 2022]. The pressure-robustness is achieved by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. An a priori error analysis proves that the velocity error is independent of the pressure and viscosity. We also propose and analyze a perturbed version of our pressure-robust EG method that allows for the elimination of the degrees of freedom corresponding to the discontinuous component of the velocity vector via static condensation. The resulting method can be viewed as a stabilized H1H^1-conforming P1\mathbb{P}_1-P0\mathbb{P}_0 method. Further, we consider efficient block preconditioners whose performances are independent of the viscosity. The theoretical results are confirmed through various numerical experiments in two and three dimensions

    Stable Lifting of Polynomial Traces on Triangles

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    We construct a right inverse of the trace operator u↩(u∣∂T,∂nu∣∂T)u \mapsto (u|_{\partial T}, \partial_n u|_{\partial T}) on the reference triangle TT that maps suitable piecewise polynomial data on ∂T\partial T into polynomials of the same degree and is bounded in all Ws,q(T)W^{s, q}(T) norms with 1<q<∞1 < q <\infty and s≄2s \geq 2. The analysis relies on new stability estimates for three classes of single edge operators. We then generalize the construction for mmth-order normal derivatives, m∈N0m \in \mathbb{N}_0.Comment: 27 pages, 1 figur

    Applications of Special Functions in High Order Finite Element Methods

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    In this thesis, we optimize different parts of high order finite element methods by application of special functions and symbolic computation. In high order finite element methods, orthogonal polynomials like the Jacobi polynomials are deeply rooted. A broad classical theory of these polynomials is known. Moreover, with modern computer algebra software we can extend this knowledge even further. Here, we apply this knowledge and software for different special functions to derive new recursive relations of local matrix entries. This massively optimizes the assembly time of local high order finite element matrices. Furthermore, the introduced algorithm is in optimal complexity. Moreover, we derive new high order dual functions, which result in fast interpolation operators. Lastly, efficient recursive algorithms for hanging node constraint matrices provided by this new dual functions are given

    Compatible finite element methods for geophysical fluid dynamics

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    This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces which are linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge-Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretising the transport terms that arise in dynamical core equation systems, and provide some example discretisations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretisations, Poisson bracket discretisations, and consistent vorticity transport.Comment: correction of some typo
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