7 research outputs found
Multilevel spectral coarsening for graph Laplacian problems with application to reservoir simulation
We extend previously developed two-level coarsening procedures for graph
Laplacian problems written in a mixed saddle point form to the fully recursive
multilevel case. The resulting hierarchy of discretizations gives rise to a
hierarchy of upscaled models, in the sense that they provide approximation in
the natural norms (in the mixed setting). This property enables us to utilize
them in three applications: (i) as an accurate reduced model, (ii) as a tool in
multilevel Monte Carlo simulations (in application to finite volume
discretizations), and (iii) for providing a sequence of nonlinear operators in
FAS (full approximation scheme) for solving nonlinear pressure equations
discretized by the conservative two-point flux approximation. We illustrate the
potential of the proposed multilevel technique in all three applications on a
number of popular benchmark problems used in reservoir simulation
Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems
This work develops a nonlinear multigrid method for diffusion problems
discretized by cell-centered finite volume methods on general unstructured
grids. The multigrid hierarchy is constructed algebraically using aggregation
of degrees of freedom and spectral decomposition of reference linear operators
associated with the aggregates. For rapid convergence, it is important that the
resulting coarse spaces have good approximation properties. In our approach,
the approximation quality can be directly improved by including more spectral
degrees of freedom in the coarsening process. Further, by exploiting local
coarsening and a piecewise-constant approximation when evaluating the nonlinear
component, the coarse level problems are assembled and solved without ever
re-visiting the fine level, an essential element for multigrid algorithms to
achieve optimal scalability. Numerical examples comparing relative performance
of the proposed nonlinear multigrid solvers with standard single-level
approaches -- Picard's and Newton's methods -- are presented. Results show that
the proposed solver consistently outperforms the single-level methods, both in
efficiency and robustness
Local Fourier analysis of multigrid for hybridized and embedded discontinuous Galerkin methods
In this paper we present a geometric multigrid method with Jacobi and Vanka
relaxation for hybridized and embedded discontinuous Galerkin discretizations
of the Laplacian. We present a local Fourier analysis (LFA) of the two-grid
error-propagation operator and show that the multigrid method applied to an
embedded discontinuous Galerkin (EDG) discretization is almost as efficient as
when applied to a continuous Galerkin discretization. We furthermore show that
multigrid applied to an EDG discretization outperforms multigrid applied to a
hybridized discontinuous Galerkin (HDG) discretization. Numerical examples
verify our LFA predictions