30 research outputs found
Legendre-Gauss-Lobatto grids and associated nested dyadic grids
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
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
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
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
By combining a certain approximation property in the spatial domain, and
weighted -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. (2017), 341-363] and [M.
Bachmayr, A. Cohen, D. D\~ung and C. Schwab, SIAM J. Numer. Anal. (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 of the approximations by them,
where 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 norm of the generating methods
where is the Gaussian probability measure on and
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
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
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