324 research outputs found
Implicit-explicit Runge–Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations
We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L 2 -energy estimates on discrete functions in physical space. Our main results are stability and quasi-optimal error estimates for smooth solutions under a standard hyperbolic CFL restriction on the time step, both in the advection-dominated and in the diffusion-dominated regimes. The theory is illustrated by numerical examples
Analysis of Compatible Discrete Operator Schemes for the Stokes Equations on Polyhedral Meshes
Compatible Discrete Operator schemes preserve basic properties of the
continuous model at the discrete level. They combine discrete differential
operators that discretize exactly topological laws and discrete Hodge operators
that approximate constitutive relations. We devise and analyze two families of
such schemes for the Stokes equations in curl formulation, with the pressure
degrees of freedom located at either mesh vertices or cells. The schemes ensure
local mass and momentum conservation. We prove discrete stability by
establishing novel discrete Poincar\'e inequalities. Using commutators related
to the consistency error, we derive error estimates with first-order
convergence rates for smooth solutions. We analyze two strategies for
discretizing the external load, so as to deliver tight error estimates when the
external load has a large irrotational or divergence-free part. Finally,
numerical results are presented on three-dimensional polyhedral meshes
Mollification in strongly Lipschitz domains with application to continuous and discrete De Rham complex
We construct mollification operators in strongly Lipschitz domains that do
not invoke non-trivial extensions, are stable for any real number
, and commute with the differential operators ,
, and . We also construct mollification
operators satisfying boundary conditions and use them to characterize the
kernel of traces related to the tangential and normal trace of vector fields.
We use the mollification operators to build projection operators onto general
-, - and -conforming
finite element spaces, with and without homogeneous boundary conditions. These
operators commute with the differential operators , ,
and , are -stable, and have optimal approximation
properties on smooth functions
A converse to Fortin's Lemma in Banach spaces
The converse of Fortin's Lemma in Banach spaces is established in this Note
Equilibrated tractions for the Hybrid High-Order method
We show how to recover equilibrated face tractions for the hybrid high-order
method for linear elasticity recently introduced in [D. A. Di Pietro and A.
Ern, A hybrid high-order locking-free method for linear elasticity on general
meshes, Comput. Meth. Appl. Mech. Engrg., 2015, 283:1-21], and prove that these
tractions are optimally convergent
A nonintrusive Reduced Basis Method applied to aeroacoustic simulations
The Reduced Basis Method can be exploited in an efficient way only if the
so-called affine dependence assumption on the operator and right-hand side of
the considered problem with respect to the parameters is satisfied. When it is
not, the Empirical Interpolation Method is usually used to recover this
assumption approximately. In both cases, the Reduced Basis Method requires to
access and modify the assembly routines of the corresponding computational
code, leading to an intrusive procedure. In this work, we derive variants of
the EIM algorithm and explain how they can be used to turn the Reduced Basis
Method into a nonintrusive procedure. We present examples of aeroacoustic
problems solved by integral equations and show how our algorithms can benefit
from the linear algebra tools available in the considered code.Comment: 28 pages, 7 figure
Variants of the Empirical Interpolation Method: symmetric formulation, choice of norms and rectangular extension
The Empirical Interpolation Method (EIM) is a greedy procedure that
constructs approximate representations of two-variable functions in separated
form. In its classical presentation, the two variables play a non-symmetric
role. In this work, we give an equivalent definition of the EIM approximation,
in which the two variables play symmetric roles. Then, we give a proof for the
existence of this approximation, and extend it up to the convergence of the
EIM, and for any norm chosen to compute the error in the greedy step. Finally,
we introduce a way to compute a separated representation in the case where the
number of selected values is different for each variable. In the case of a
physical field measured by sensors, this is useful to discard a broken sensor
while keeping the information provided by the associated selected field.Comment: 7 page
Accurate and efficient evaluation of the a posteriori error estimator in the reduced basis method
The reduced basis method is a model reduction technique yielding substantial
savings of computational time when a solution to a parametrized equation has to
be computed for many values of the parameter. Certification of the
approximation is possible by means of an a posteriori error bound. Under
appropriate assumptions, this error bound is computed with an algorithm of
complexity independent of the size of the full problem. In practice, the
evaluation of the error bound can become very sensitive to round-off errors. We
propose herein an explanation of this fact. A first remedy has been proposed in
[F. Casenave, Accurate \textit{a posteriori} error evaluation in the reduced
basis method. \textit{C. R. Math. Acad. Sci. Paris} \textbf{350} (2012)
539--542.]. Herein, we improve this remedy by proposing a new approximation of
the error bound using the Empirical Interpolation Method (EIM). This method
achieves higher levels of accuracy and requires potentially less
precomputations than the usual formula. A version of the EIM stabilized with
respect to round-off errors is also derived. The method is illustrated on a
simple one-dimensional diffusion problem and a three-dimensional acoustic
scattering problem solved by a boundary element method.Comment: 26 pages, 10 figures. ESAIM: Mathematical Modelling and Numerical
Analysis, 201
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