72 research outputs found
Optimized Schwarz waveform relaxation and discontinuous Galerkin time stepping for heterogeneous problems
We design and analyze a Schwarz waveform relaxation algorithm for domain
decomposition of advection-diffusion-reaction problems with strong
heterogeneities. The interfaces are curved, and we use optimized Robin or
Ventcell transmission conditions. We analyze the semi-discretization in time
with Discontinuous Galerkin as well. We also show two-dimensional numerical
results using generalized mortar finite elements in space
An efficient way to assemble finite element matrices in vector languages
Efficient Matlab codes in 2D and 3D have been proposed recently to assemble
finite element matrices. In this paper we present simple, compact and efficient
vectorized algorithms, which are variants of these codes, in arbitrary
dimension, without the use of any lower level language. They can be easily
implemented in many vector languages (e.g. Matlab, Octave, Python, Scilab, R,
Julia, C++ with STL,...). The principle of these techniques is general, we
present it for the assembly of several finite element matrices in arbitrary
dimension, in the P1 finite element case. We also provide an extension of the
algorithms to the case of a system of PDE's. Then we give an extension to
piecewise polynomials of higher order. We compare numerically the performance
of these algorithms in Matlab, Octave and Python, with that in FreeFEM++ and in
a compiled language such as C. Examples show that, unlike what is commonly
believed, the performance is not radically worse than that of C : in the
best/worst cases, selected vector languages are respectively 2.3/3.5 and
2.9/4.1 times slower than C in the scalar and vector cases. We also present
numerical results which illustrate the computational costs of these algorithms
compared to standard algorithms and to other recent ones
Robin Schwarz algorithm for the NICEM Method: the Pq finite element case
In Gander et al. [2004] we proposed a new non-conforming domain decomposition
paradigm, the New Interface Cement Equilibrated Mortar (NICEM) method, based on
Schwarz type methods that allows for the use of Robin interface conditions on
non-conforming grids. The error analysis was done for P1 finite elements, in 2D
and 3D. In this paper, we provide new numerical analysis results that allow to
extend this error analysis in 2D for piecewise polynomials of higher order and
also prove the convergence of the iterative algorithm in all these cases.Comment: arXiv admin note: substantial text overlap with arXiv:0705.028
Space-time domain decomposition for advection-diffusion problems in mixed formulations
This paper is concerned with the numerical solution of porous-media flow and
transport problems , i. e. heterogeneous, advection-diffusion problems. Its aim
is to investigate numerical schemes for these problems in which different time
steps can be used in different parts of the domain. Global-in-time,
non-overlapping domain-decomposition methods are coupled with operator
splitting making possible the different treatment of the advection and
diffusion terms. Two domain-decomposition methods are considered: one uses the
time-dependent Steklov--Poincar{\'e} operator and the other uses optimized
Schwarz waveform relaxation (OSWR) based on Robin transmission conditions. For
each method, a mixed formulation of an interface problem on the space-time
interface is derived, and different time grids are employed to adapt to
different time scales in the subdomains. A generalized Neumann-Neumann
preconditioner is proposed for the first method. To illustrate the two methods
numerical results for two-dimensional problems with strong heterogeneities are
presented. These include both academic problems and more realistic prototypes
for simulations for the underground storage of nuclear waste
A new Cement to Glue non-conforming Grids with Robin interface conditions: the finite element case
We design and analyze a new non-conforming domain decomposition method based on Schwarz type approaches that allows for the use of Robin interface conditions on non-conforming grids. The method is proven to be well posed, and the iterative solver to converge. The error analysis is performed in 2D piecewise polynomials of low and high order and extended in 3D for elements. Numerical results in 2D illustrate the new method
A Simple and Efficient Tool for Trapping Gravid Anopheles at Breeding Sites.
No effective tool currently exists for trapping ovipositing malaria vectors. This creates a gap in our ability to investigate the behavior and ecology of gravid Anopheles.\ud
Here we describe a simple trap that collects ovipositing Anopheline and Culicine mosquitoes. It consists of an acetate sheet coated in glue that floats on the water surface. Ten breeding sites were selected in rural Tanzania and 10 sticky traps set in each. These caught a total of 74 gravid Anopheles (54 An. arabiensis, 1 An. gambiae s.s. and 16 unamplified) and 1333 gravid Culicines, in just two trap nights. This simple sampling tool provides an opportunity to further our understanding of the behavior and ecology of gravid female Anophelines. It strongly implies that at least two of the major vectors of malaria in Africa land on the water surface during the oviposition process, and demonstrates that Anophelines and Culicines often share the same breeding sites. This simple and efficient trap has clear potential for the study of oviposition site choice and productivity, gravid dispersal, and vector control techniques which use oviposition behavior as a means of disseminating larvicides
Space-time Domain Decomposition and Mixed Formulation for solving reduced fracture models
International audienceIn this paper we are interested in the "fast path" fracture and we aim to use global-in-time, nonoverlapping domain decomposition methods to model flow and transport problems in a porous medium containing such a fracture. We consider a reduced model in which the fracture is treated as an interface between the two subdomains. Two domain decomposition methods are considered: one uses the time-dependent SteklovPoincaré operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Ventcell transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interface is derived, and different time grids are employed to adapt to different time scales in the subdomains and in the fracture. Demonstrations of the well-posedness of the Ventcell subdomain problems is given for the mixed formulation. An analysis for the convergence factor of the OSWR algorithm is given in the case with fractures to compute the optimized parameters. Numerical results for two-dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods
Space-time domain decomposition for mixed formulations of diffusion equations
International audienceThe far field simulation of underground nuclear waste disposal site requires a high computational cost due to the widely varying properties of different materials, the different length and time scales, and the high accuracy requirements. Nonoverlapping domain decomposition methods allow local adaptation in both space and time and result in parallel algorithms. We have extended the optimized Schwarz waveform relaxation (OSWR) method, successfully used for finite elements and finite volumes, to the case of mixed finite elements with their local mass-conservation property. Another choice is the substructuring method, which has been shown to be efficient for steady state problems with strong heterogeneities. We study a time-dependent Schur complement method, which is the algebraic counterpart of the discrete Steklov Poincaré operator, and introduce the Neumann preconditioner as well as weight matrices designed to make the convergence speed independent of the heterogeneities. Both methods enable the use of local time steps when the subdomains have highly different physical properties. Their performance is illustrated on test cases suggested by nuclear waste disposal problems
Efficient interface conditions for the coupling of ocean models
International audienceAlthough the two-way nesting is the most usual coupling method in the ocean community, it does not address the correct problem but an approximation, and does not ensure enough regularity through the interface between the two models1. The correct approach consisting in a full coupling is more difficult and expensive than the two-way nesting, since it requires to find and implement an algorithm ensuring that the solutions in each domain satisfy the regularity conditions through the interface. The global-in-time non-overlapping Schwarz algorithm is particularly well suited for such a coupling, and can lead to improved physical results. Our work thus aims at improving the ocean coupling by determining efficient interface conditions for the usual ocean equations (the so-called 3-D primitive equations). These ones are composed of advection-diffusion equations for tracers such as temperature and salinity, and dynamics equations, which can be approximated in the 2-D case by the shallow-water equations
Space-Time Domain Decomposition Methods for Diffusion Problems in Mixed Formulations
This paper is concerned with global-in-time, nonoverlapping domain
decomposition methods for the mixed formulation of the diffusion problem. Two
approaches are considered: one uses the time-dependent Steklov-Poincar\'e
operator and the other uses Optimized Schwarz Waveform Relaxation (OSWR) based
on Robin transmission conditions. For each method, a mixed formulation of an
interface problem on the space-time interfaces between subdomains is derived,
and different time grids are employed to adapt to different time scales in the
subdomains. Demonstrations of the well-posedness of the subdomain problems
involved in each method and a convergence proof of the OSWR algorithm are given
for the mixed formulation. Numerical results for 2D problems with strong
heterogeneities are presented to illustrate the performance of the two methods
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