1,988 research outputs found
Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations
Computational costs of numerically solving multidimensional partial
differential equations (PDEs) increase significantly when the spatial
dimensions of the PDEs are high, due to large number of spatial grid points.
For multidimensional reaction-diffusion equations, stiffness of the system
provides additional challenges for achieving efficient numerical simulations.
In this paper, we propose a class of Krylov implicit integration factor (IIF)
discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion
equations on high spatial dimensions. The key ingredient of spatial DG
discretization is the multiwavelet bases on nested sparse grids, which can
significantly reduce the numbers of degrees of freedom. To deal with the
stiffness of the DG spatial operator in discretizing reaction-diffusion
equations, we apply the efficient IIF time discretization methods, which are a
class of exponential integrators. Krylov subspace approximations are used to
evaluate the large size matrix exponentials resulting from IIF schemes for
solving PDEs on high spatial dimensions. Stability and error analysis for the
semi-discrete scheme are performed. Numerical examples of both scalar equations
and systems in two and three spatial dimensions are provided to demonstrate the
accuracy and efficiency of the methods. The stiffness of the reaction-diffusion
equations is resolved well and large time step size computations are obtained
h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems
In this work we exploit agglomeration based -multigrid preconditioners to
speed-up the iterative solution of discontinuous Galerkin discretizations of
the Stokes and Navier-Stokes equations. As a distinctive feature -coarsened
mesh sequences are generated by recursive agglomeration of a fine grid,
admitting arbitrarily unstructured grids of complex domains, and agglomeration
based discontinuous Galerkin discretizations are employed to deal with
agglomerated elements of coarse levels. Both the expense of building coarse
grid operators and the performance of the resulting multigrid iteration are
investigated. For the sake of efficiency coarse grid operators are inherited
through element-by-element projections, avoiding the cost of numerical
integration over agglomerated elements. Specific care is devoted to the
projection of viscous terms discretized by means of the BR2 dG method. We
demonstrate that enforcing the correct amount of stabilization on coarse grids
levels is mandatory for achieving uniform convergence with respect to the
number of levels. The numerical solution of steady and unsteady, linear and
non-linear problems is considered tackling challenging 2D test cases and 3D
real life computations on parallel architectures. Significant execution time
gains are documented.Comment: 78 pages, 7 figure
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
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