496 research outputs found
A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure
In this paper we formulate and test numerically a fully-coupled discontinuous
Galerkin (DG) method for incompressible two-phase flow with discontinuous
capillary pressure. The spatial discretization uses the symmetric interior
penalty DG formulation with weighted averages and is based on a wetting-phase
potential / capillary potential formulation of the two-phase flow system. After
discretizing in time with diagonally implicit Runge-Kutta schemes the resulting
systems of nonlinear algebraic equations are solved with Newton's method and
the arising systems of linear equations are solved efficiently and in parallel
with an algebraic multigrid method. The new scheme is investigated for various
test problems from the literature and is also compared to a cell-centered
finite volume scheme in terms of accuracy and time to solution. We find that
the method is accurate, robust and efficient. In particular no post-processing
of the DG velocity field is necessary in contrast to results reported by
several authors for decoupled schemes. Moreover, the solver scales well in
parallel and three-dimensional problems with up to nearly 100 million degrees
of freedom per time step have been computed on 1000 processors
Efficient operator-coarsening multigrid schemes for local discontinuous Galerkin methods
An efficient -multigrid scheme is presented for local discontinuous
Galerkin (LDG) discretizations of elliptic problems, formulated around the idea
of separately coarsening the underlying discrete gradient and divergence
operators. We show that traditional multigrid coarsening of the primal
formulation leads to poor and suboptimal multigrid performance, whereas
coarsening of the flux formulation leads to optimal convergence and is
equivalent to a purely geometric multigrid method. The resulting
operator-coarsening schemes do not require the entire mesh hierarchy to be
explicitly built, thereby obviating the need to compute quadrature rules,
lifting operators, and other mesh-related quantities on coarse meshes. We show
that good multigrid convergence rates are achieved in a variety of numerical
tests on 2D and 3D uniform and adaptive Cartesian grids, as well as for curved
domains using implicitly defined meshes and for multi-phase elliptic interface
problems with complex geometry. Extension to non-LDG discretizations is briefly
discussed
Hybrid multigrid methods for high-order discontinuous Galerkin discretizations
The present work develops hybrid multigrid methods for high-order
discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free
operator evaluation on tensor product elements is used to devise a
computationally efficient PDE solver. The multigrid hierarchy exploits all
possibilities of geometric, polynomial, and algebraic coarsening, targeting
engineering applications on complex geometries. Additionally, a transfer from
discontinuous to continuous function spaces is performed within the multigrid
hierarchy. This does not only further reduce the problem size of the
coarse-grid problem, but also leads to a discretization most suitable for
state-of-the-art algebraic multigrid methods applied as coarse-grid solver. The
relevant design choices regarding the selection of optimal multigrid coarsening
strategies among the various possibilities are discussed with the metric of
computational costs as the driving force for algorithmic selections. We find
that a transfer to a continuous function space at highest polynomial degree (or
on the finest mesh), followed by polynomial and geometric coarsening, shows the
best overall performance. The success of this particular multigrid strategy is
due to a significant reduction in iteration counts as compared to a transfer
from discontinuous to continuous function spaces at lowest polynomial degree
(or on the coarsest mesh). The coarsening strategy with transfer to a
continuous function space on the finest level leads to a multigrid algorithm
that is robust with respect to the penalty parameter of the SIPG method.
Detailed numerical investigations are conducted for a series of examples
ranging from academic test cases to more complex, practically relevant
geometries. Performance comparisons to state-of-the-art methods from the
literature demonstrate the versatility and computational efficiency of the
proposed multigrid algorithms
A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems
We propose and analyze a two-level method for mimetic finite difference
approximations of second order elliptic boundary value problems. We prove that
the two-level algorithm is uniformly convergent, i.e., the number of iterations
needed to achieve convergence is uniformly bounded independently of the
characteristic size of the underling partition. We also show that the resulting
scheme provides a uniform preconditioner with respect to the number of degrees
of freedom. Numerical results that validate the theory are also presented
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