1,160 research outputs found
Degenerate anisotropic elliptic problems and magnetized plasma simulations
This paper is devoted to the numerical approximation of a degenerate
anisotropic elliptic problem. The numerical method is designed for arbitrary
space-dependent anisotropy directions and does not require any specially
adapted coordinate system. It is also designed to be equally accurate in the
strongly and the mildly anisotropic cases. The method is applied to the
Euler-Lorentz system, in the drift-fluid limit. This system provides a model
for magnetized plasmas
Multiscale methods for problems with complex geometry
We propose a multiscale method for elliptic problems on complex domains, e.g.
domains with cracks or complicated boundary. For local singularities this paper
also offers a discrete alternative to enrichment techniques such as XFEM. We
construct corrected coarse test and trail spaces which takes the fine scale
features of the computational domain into account. The corrections only need to
be computed in regions surrounding fine scale geometric features. We achieve
linear convergence rate in energy norm for the multiscale solution. Moreover,
the conditioning of the resulting matrices is not affected by the way the
domain boundary cuts the coarse elements in the background mesh. The analytical
findings are verified in a series of numerical experiments
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
Proceedings for the ICASE Workshop on Heterogeneous Boundary Conditions
Domain Decomposition is a complex problem with many interesting aspects. The choice of decomposition can be made based on many different criteria, and the choice of interface of internal boundary conditions are numerous. The various regions under study may have different dynamical balances, indicating that different physical processes are dominating the flow in these regions. This conference was called in recognition of the need to more clearly define the nature of these complex problems. This proceedings is a collection of the presentations and the discussion groups
Isogeometric Analysis on V-reps: first results
Inspired by the introduction of Volumetric Modeling via volumetric
representations (V-reps) by Massarwi and Elber in 2016, in this paper we
present a novel approach for the construction of isogeometric numerical methods
for elliptic PDEs on trimmed geometries, seen as a special class of more
general V-reps. We develop tools for approximation and local re-parametrization
of trimmed elements for three dimensional problems, and we provide a
theoretical framework that fully justify our algorithmic choices. We validate
our approach both on two and three dimensional problems, for diffusion and
linear elasticity.Comment: 36 pages, 44 figures. Reviewed versio
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
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