30 research outputs found

    Legendre-Gauss-Lobatto grids and associated nested dyadic grids

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    Legendre-Gauss-Lobatto (LGL) grids play a pivotal role in nodal spectral methods for the numerical solution of partial differential equations. They not only provide efficient high-order quadrature rules, but give also rise to norm equivalences that could eventually lead to efficient preconditioning techniques in high-order methods. Unfortunately, a serious obstruction to fully exploiting the potential of such concepts is the fact that LGL grids of different degree are not nested. This affects, on the one hand, the choice and analysis of suitable auxiliary spaces, when applying the auxiliary space method as a principal preconditioning paradigm, and, on the other hand, the efficient solution of the auxiliary problems. As a central remedy, we consider certain nested hierarchies of dyadic grids of locally comparable mesh size, that are in a certain sense properly associated with the LGL grids. Their actual suitability requires a subtle analysis of such grids which, in turn, relies on a number of refined properties of LGL grids. The central objective of this paper is to derive just these properties. This requires first revisiting properties of close relatives to LGL grids which are subsequently used to develop a refined analysis of LGL grids. These results allow us then to derive the relevant properties of the associated dyadic grids.Comment: 35 pages, 7 figures, 2 tables, 2 algorithms; Keywords: Legendre-Gauss-Lobatto grid, dyadic grid, graded grid, nested grid

    Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes

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    This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral Discontinuous Galerkin discretizations of elliptic boundary value problems. While presently known techniques realize a growth of the condition numbers that is logarithmic in the polynomial degrees when all degrees are equal and quadratic otherwise, our main objective is to realize full robustness with respect to arbitrarily large locally varying polynomial degrees degrees, i.e., under mild grading constraints condition numbers stay uniformly bounded with respect to the mesh size and variable degrees. The conceptual foundation of the envisaged preconditioners is the auxiliary space method. The main conceptual ingredients that will be shown in this framework to yield "optimal" preconditioners in the above sense are Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic nested dyadic grids as well as specially adapted wavelet preconditioners for the resulting low order auxiliary problems. Moreover, the preconditioners have a modular form that facilitates somewhat simplified partial realizations. One of the components can, for instance, be conveniently combined with domain decomposition, at the expense though of a logarithmic growth of condition numbers. Our analysis is complemented by quantitative experimental studies of the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents for better readability, part on wavelet preconditioner adde

    Entropy Stable Staggered Grid Spectral Collocation for the Burgers' and Compressible Navier-Stokes Equations

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    Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for Burgers' and the compressible Navier-Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [1, 2], extends the applicable set of points from tensor product, Legendre-Gauss-Lobatto (LGL) to a combination of tensor product Legendre-Gauss (LG) and LGL points. The new semi-discrete operators discretely conserve mass, momentum, energy and satisfy a mathematical entropy inequality for both Burgers' and the compressible Navier-Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG operators are significantly more accurate, although more costly to implement. The LG and LGL operators exhibit similar robustness, as is demonstrated using test problems known to be problematic for operators that lack a nonlinearly stability proof for the compressible Navier-Stokes equations (e.g., discontinuous Galerkin, spectral difference, or flux reconstruction operators)

    Towards an Entropy Stable Spectral Element Framework for Computational Fluid Dynamics

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    Entropy stable (SS) discontinuous spectral collocation formulations of any order are developed for the compressible Navier-Stokes equations on hexahedral elements. Recent progress on two complementary efforts is presented. The first effort is a generalization of previous SS spectral collocation work to extend the applicable set of points from tensor product, Legendre-Gauss-Lobatto (LGL) to tensor product Legendre-Gauss (LG) points. The LG and LGL point formulations are compared on a series of test problems. Although being more costly to implement, it is shown that the LG operators are significantly more accurate on comparable grids. Both the LGL and LG operators are of comparable efficiency and robustness, as is demonstrated using test problems for which conventional FEM techniques suffer instability. The second effort generalizes previous SS work to include the possibility of p-refinement at non-conforming interfaces. A generalization of existing entropy stability machinery is developed to accommodate the nuances of fully multi-dimensional summation-by-parts (SBP) operators. The entropy stability of the compressible Euler equations on non-conforming interfaces is demonstrated using the newly developed LG operators and multi-dimensional interface interpolation operators

    Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

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    By combining a certain approximation property in the spatial domain, and weighted ℓ2\ell_2-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in [M. Bachmayr, A. Cohen, R. DeVore and G. Migliorati, ESAIM Math. Model. Numer. Anal. 51\bf 51(2017), 341-363] and [M. Bachmayr, A. Cohen, D. D\~ung and C. Schwab, SIAM J. Numer. Anal. 55\bf 55(2017), 2151-2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We explicitly construct such methods and prove corresponding convergence rates in nn of the approximations by them, where nn is a number characterizing computation complexity. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods. Moreover, they generate in a natural way discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the corresponding integration can be estimated via the error in the Bochner space L1(R∞,V,γ)L_1({\mathbb R}^\infty,V,\gamma) norm of the generating methods where γ\gamma is the Gaussian probability measure on R∞{\mathbb R}^\infty and VV is the energy space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and by-product problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rate of non-fully discrete obtained in this paper improves the known one

    Three-dimensional modelling and inversion of controlled source electromagnetic data

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    The marine Controlled Source Electromagnetic (CSEM) method is an important and almost self-contained discipline in the toolkit of methods used by geophysicists for probing the earth. It has increasingly attracted attention from industry during the past decade due to its potential in detecting valuable natural resources such as oil and gas. A method for three-dimensional CSEM modelling in the frequency domain is presented. The electric field is decomposed in primary and secondary components, as this leads to a more stable solution near the source position. The primary field is computed using a resistivity model for which a closed form of solution exists, for example a homogeneous or layered resistivity model. The secondary electric field is computed by discretizing a second order partial differential equation for the electric field, also referred in the literature as the vector Helmholtz equation, using the edge finite element method. A range of methods for the solution of the linear system derived from the edge finite element discretization are investigated. The magnetic field is computed subsequently, from the solution for the electric field, using a local finite difference approximation of Faraday’s law and an interpolation method. Tests, that compare the solution obtained using the presented method with the solution computed using alternative codes for 1D and 3D synthetic models, show that the implemented approach is suitable for CSEM forward modelling and is an alternative to existing codes. An algorithm for 3D inversion of CSEM data in the frequency domain was developed and implemented. The inverse problem is solved using the L-BFGS method and is regularized with a smoothing constraint. The inversion algorithm uses the presented forward modelling scheme for the computation of the field responses and the adjoint field for the computation of the gradient of the misfit function. The presented algorithm was tested for a synthetic example, showing that it is capable of reconstructing a resistivity model which fits the synthetic data and is close to the original resistivity model in the least-squares sense. Inversion of CSEM data is known to lead to images with low spatial resolution. It is well known that integration with complementary data sets mitigates this problem. It is presented an algorithm for the integration of an acoustic velocity model, which is known a priori, in the inversion scheme. The algorithm was tested in a synthetic example and the results demonstrate that the presented methodology is promising for the improvement of resistivity models obtained from CSEM data

    Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

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    This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions
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