39,323 research outputs found
Convergence and Optimality of Adaptive Mixed Finite Element Methods
The convergence and optimality of adaptive mixed finite element methods for
the Poisson equation are established in this paper. The main difficulty for
mixed finite element methods is the lack of minimization principle and thus the
failure of orthogonality. A quasi-orthogonality property is proved using the
fact that the error is orthogonal to the divergence free subspace, while the
part of the error that is not divergence free can be bounded by the data
oscillation using a discrete stability result. This discrete stability result
is also used to get a localized discrete upper bound which is crucial for the
proof of the optimality of the adaptive approximation
Adaptive vertex-centered finite volume methods for general second-order linear elliptic PDEs
We prove optimal convergence rates for the discretization of a general
second-order linear elliptic PDE with an adaptive vertex-centered finite volume
scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54
(2016), pp. 2228--2255] was restricted to symmetric problems, the present
analysis also covers non-symmetric problems and hence the important case of
present convection
Adaptive, Anisotropic and Hierarchical cones of Discrete Convex functions
We address the discretization of optimization problems posed on the cone of
convex functions, motivated in particular by the principal agent problem in
economics, which models the impact of monopoly on product quality. Consider a
two dimensional domain, sampled on a grid of N points. We show that the cone of
restrictions to the grid of convex functions is in general characterized by N^2
linear inequalities; a direct computational use of this description therefore
has a prohibitive complexity. We thus introduce a hierarchy of sub-cones of
discrete convex functions, associated to stencils which can be adaptively,
locally, and anisotropically refined. Numerical experiments optimize the
accuracy/complexity tradeoff through the use of a-posteriori stencil refinement
strategies.Comment: 35 pages, 11 figures. (Second version fixes a small bug in Lemma 3.2.
Modifications are anecdotic.
A scalable parallel finite element framework for growing geometries. Application to metal additive manufacturing
This work introduces an innovative parallel, fully-distributed finite element
framework for growing geometries and its application to metal additive
manufacturing. It is well-known that virtual part design and qualification in
additive manufacturing requires highly-accurate multiscale and multiphysics
analyses. Only high performance computing tools are able to handle such
complexity in time frames compatible with time-to-market. However, efficiency,
without loss of accuracy, has rarely held the centre stage in the numerical
community. Here, in contrast, the framework is designed to adequately exploit
the resources of high-end distributed-memory machines. It is grounded on three
building blocks: (1) Hierarchical adaptive mesh refinement with octree-based
meshes; (2) a parallel strategy to model the growth of the geometry; (3)
state-of-the-art parallel iterative linear solvers. Computational experiments
consider the heat transfer analysis at the part scale of the printing process
by powder-bed technologies. After verification against a 3D benchmark, a
strong-scaling analysis assesses performance and identifies major sources of
parallel overhead. A third numerical example examines the efficiency and
robustness of (2) in a curved 3D shape. Unprecedented parallelism and
scalability were achieved in this work. Hence, this framework contributes to
take on higher complexity and/or accuracy, not only of part-scale simulations
of metal or polymer additive manufacturing, but also in welding, sedimentation,
atherosclerosis, or any other physical problem where the physical domain of
interest grows in time
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