313 research outputs found
Error estimates for stabilized finite element methods applied to ill-posed problems
We propose an analysis for the stabilized finite element methods proposed in,
E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and
ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput., 35(6)
2013, valid in the case of ill-posed problems for which only weak continuous
dependence can be assumed. A priori and a posteriori error estimates are
obtained without assuming coercivity or inf-sup stability of the continuous
problem. A numerical example illustrates the theory.Comment: The theoretical part is submitted to Comptes Rendus Mathematiques and
the numerical example is taken from the reference mentioned in the abstrac
Stabilised finite element methods for ill-posed problems with conditional stability
In this paper we discuss the adjoint stabilised finite element method
introduced in, E. Burman, Stabilized finite element methods for nonsymmetric,
noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on
Scientific Computing, and how it may be used for the computation of solutions
to problems for which the standard stability theory given by the Lax-Milgram
Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to
ill-posed problems that have some conditional stability property and prove
(conditional) error estimates in an abstract framework. As a model problem we
consider the elliptic Cauchy problem and provide a complete numerical analysis
for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building
Bridges: Connections and Challenges in Modern Approaches to Numerical Partial
Differential Equation
Robust error estimates in weak norms for advection dominated transport problems with rough data
We consider mixing problems in the form of transient convection--diffusion
equations with a velocity vector field with multiscale character and rough
data. We assume that the velocity field has two scales, a coarse scale with
slow spatial variation, which is responsible for advective transport and a fine
scale with small amplitude that contributes to the mixing. For this problem we
consider the estimation of filtered error quantities for solutions computed
using a finite element method with symmetric stabilization. A posteriori error
estimates and a priori error estimates are derived using the multiscale
decomposition of the advective velocity to improve stability. All estimates are
independent both of the P\'eclet number and of the regularity of the exact
solution
Implicit-explicit RungeāKutta schemes and finite elements with symmetric stabilization for advection-diffusion equations
We analyze a two-stage implicit-explicit RungeāKutta 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
Stabilized nonconforming finite element methods for data assimilation in incompressible flows
We consider a stabilized nonconforming finite element method for data
assimilation in incompressible flow subject to the Stokes' equations. The
method uses a primal dual structure that allows for the inclusion of
nonstandard data. Error estimates are obtained that are optimal compared to the
conditional stability of the ill-posed data assimilation problem
Primal dual mixed finite element methods for indefinite advection--diffusion equations
We consider primal-dual mixed finite element methods for the
advection--diffusion equation. For the primal variable we use standard
continuous finite element space and for the flux we use the Raviart-Thomas
space. We prove optimal a priori error estimates in the energy- and the
-norms for the primal variable in the low Peclet regime. In the high
Peclet regime we also prove optimal error estimates for the primal variable in
the norm for smooth solutions. Numerically we observe that the method
eliminates the spurious oscillations close to interior layers that pollute the
solution of the standard Galerkin method when the local Peclet number is high.
This method, however, does produce spurious solutions when outflow boundary
layer presents. In the last section we propose two simple strategies to remove
such numerical artefacts caused by the outflow boundary layer and validate them
numerically.Comment: 25 pages, 6 figures, 5 table
A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime
The numerical approximation of an inverse problem subject to the
convection--diffusion equation when diffusion dominates is studied. We derive
Carleman estimates that are on a form suitable for use in numerical analysis
and with explicit dependence on the P\'eclet number. A stabilized finite
element method is then proposed and analysed. An upper bound on the condition
number is first derived. Combining the stability estimates on the continuous
problem with the numerical stability of the method, we then obtain error
estimates in local - or -norms that are optimal with respect to the
approximation order, the problem's stability and perturbations in data. The
convergence order is the same for both norms, but the -estimate requires
an additional divergence assumption for the convective field. The theory is
illustrated in some computational examples.Comment: 21 pages, 6 figures; in v2 we added two remarks and an appendix on
psiDOs, and made some minor correction
A finite element data assimilation method for the wave equation
We design a primal-dual stabilized finite element method for the numerical
approximation of a data assimilation problem subject to the acoustic wave
equation. For the forward problem, piecewise affine, continuous, finite element
functions are used for the approximation in space and backward differentiation
is used in time. Stabilizing terms are added on the discrete level. The design
of these terms is driven by numerical stability and the stability of the
continuous problem, with the objective of minimizing the computational error.
Error estimates are then derived that are optimal with respect to the
approximation properties of the numerical scheme and the stability properties
of the continuous problem. The effects of discretizing the (smooth) domain
boundary and other perturbations in data are included in the analysis.Comment: 23 page
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