6,451 research outputs found
Efficient Resolution of Anisotropic Structures
We highlight some recent new delevelopments concerning the sparse
representation of possibly high-dimensional functions exhibiting strong
anisotropic features and low regularity in isotropic Sobolev or Besov scales.
Specifically, we focus on the solution of transport equations which exhibit
propagation of singularities where, additionally, high-dimensionality enters
when the convection field, and hence the solutions, depend on parameters
varying over some compact set. Important constituents of our approach are
directionally adaptive discretization concepts motivated by compactly supported
shearlet systems, and well-conditioned stable variational formulations that
support trial spaces with anisotropic refinements with arbitrary
directionalities. We prove that they provide tight error-residual relations
which are used to contrive rigorously founded adaptive refinement schemes which
converge in . Moreover, in the context of parameter dependent problems we
discuss two approaches serving different purposes and working under different
regularity assumptions. For frequent query problems, making essential use of
the novel well-conditioned variational formulations, a new Reduced Basis Method
is outlined which exhibits a certain rate-optimal performance for indefinite,
unsymmetric or singularly perturbed problems. For the radiative transfer
problem with scattering a sparse tensor method is presented which mitigates or
even overcomes the curse of dimensionality under suitable (so far still
isotropic) regularity assumptions. Numerical examples for both methods
illustrate the theoretical findings
A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension
Finite volume methods for problems involving second order operators with full
diffusion matrix can be used thanks to the definition of a discrete gradient
for piecewise constant functions on unstructured meshes satisfying an
orthogonality condition. This discrete gradient is shown to satisfy a strong
convergence property on the interpolation of regular functions, and a weak one
on functions bounded for a discrete norm. To highlight the importance of
both properties, the convergence of the finite volume scheme on a homogeneous
Dirichlet problem with full diffusion matrix is proven, and an error estimate
is provided. Numerical tests show the actual accuracy of the method
Adjoint-Based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods
We present a robust and efficient target-based mesh adaptation methodology,
building on hybridized discontinuous Galerkin schemes for (nonlinear)
convection-diffusion problems, including the compressible Euler and
Navier-Stokes equations. Hybridization of finite element discretizations has
the main advantage, that the resulting set of algebraic equations has globally
coupled degrees of freedom only on the skeleton of the computational mesh.
Consequently, solving for these degrees of freedom involves the solution of a
potentially much smaller system. This not only reduces storage requirements,
but also allows for a faster solution with iterative solvers. The mesh
adaptation is driven by an error estimate obtained via a discrete adjoint
approach. Furthermore, the computed target functional can be corrected with
this error estimate to obtain an even more accurate value. The aim of this
paper is twofold: Firstly, to show the superiority of adjoint-based mesh
adaptation over uniform and residual-based mesh refinement, and secondly to
investigate the efficiency of the global error estimate
On discrete functional inequalities for some finite volume schemes
We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincar\'e-Sobolev
inequalities for some approximations with arbitrary boundary values on finite
volume meshes. The keypoint of our approach is to use the continuous embedding
of the space into for a Lipschitz domain , with . Finally, we give several
applications to discrete duality finite volume (DDFV) schemes which are used
for the approximation of nonlinear and non isotropic elliptic and parabolic
problems
A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow
We present a comparison between hybridized and non-hybridized discontinuous
Galerkin methods in the context of target-based hp-adaptation for compressible
flow problems. The aim is to provide a critical assessment of the computational
efficiency of hybridized DG methods. Hybridization of finite element
discretizations has the main advantage, that the resulting set of algebraic
equations has globally coupled degrees of freedom only on the skeleton of the
computational mesh. Consequently, solving for these degrees of freedom involves
the solution of a potentially much smaller system. This not only reduces
storage requirements, but also allows for a faster solution with iterative
solvers. Using a discrete-adjoint approach, sensitivities with respect to
output functionals are computed to drive the adaptation. From the error
distribution given by the adjoint-based error estimator, h- or p-refinement is
chosen based on the smoothness of the solution which can be quantified by
properly-chosen smoothness indicators. Numerical results are shown for
subsonic, transonic, and supersonic flow around the NACA0012 airfoil.
hp-adaptation proves to be superior to pure h-adaptation if discontinuous or
singular flow features are involved. In all cases, a higher polynomial degree
turns out to be beneficial. We show that for polynomial degree of approximation
p=2 and higher, and for a broad range of test cases, HDG performs better than
DG in terms of runtime and memory requirements
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