2,558 research outputs found
Data assimilation for the heat equation using stabilized finite element methods
We consider data assimilation for the heat equation using a finite element space semi-discretization. The approach is optimization based, but the design of regularization operators and parameters rely on techniques from the theory of stabilized finite elements. The space semi-discretized system is shown to admit a unique solution. Combining sharp estimates of the numerical stability of the discrete scheme and conditional stability estimates of the ill-posed continuous pde-model we then derive error estimates that reflect the approximation order of the finite element space and the stability of the continuous model. Two different data assimilation situations with different stability properties are considered to illustrate the framework. Full detail on how to adapt known stability estimates for the continuous model to work with the numerical analysis framework is given in âAppendixâ
Fully discrete finite element data assimilation method for the heat equation
We consider a finite element discretization for the reconstruction of the
final state of the heat equation, when the initial data is unknown, but
additional data is given in a sub domain in the space time. For the
discretization in space we consider standard continuous affine finite element
approximation, and the time derivative is discretized using a backward
differentiation. We regularize the discrete system by adding a penalty of the
-semi-norm of the initial data, scaled with the mesh-parameter. The
analysis of the method uses techniques developed in E. Burman and L. Oksanen,
Data assimilation for the heat equation using stabilized finite element
methods, arXiv, 2016, combining discrete stability of the numerical method with
sharp Carleman estimates for the physical problem, to derive optimal error
estimates for the approximate solution. For the natural space time energy norm,
away from , the convergence is the same as for the classical problem with
known initial data, but contrary to the classical case, we do not obtain faster
convergence for the -norm at the final time
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
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
Unique continuation for the Helmholtz equation using stabilized finite element methods
In this work we consider the computational approximation of a unique
continuation problem for the Helmholtz equation using a stabilized finite
element method. First conditional stability estimates are derived for which,
under a convexity assumption on the geometry, the constants grow at most
linearly in the wave number. Then these estimates are used to obtain error
bounds for the finite element method that are explicit with respect to the wave
number. Some numerical illustrations are given.Comment: corrected typos; included suggestions from reviewer
Space time stabilized finite element methods for a unique continuation problem subject to the wave equation
We consider a stabilized finite element method based on a spacetime
formulation, where the equations are solved on a global (unstructured)
spacetime mesh. A unique continuation problem for the wave equation is
considered, where data is known in an interior subset of spacetime. For this
problem, we consider a primal-dual discrete formulation of the continuum
problem with the addition of stabilization terms that are designed with the
goal of minimizing the numerical errors. We prove error estimates using the
stability properties of the numerical scheme and a continuum observability
estimate, based on the sharp geometric control condition by Bardos, Lebeau and
Rauch. The order of convergence for our numerical scheme is optimal with
respect to stability properties of the continuum problem and the interpolation
errors of approximating with polynomial spaces. Numerical examples are provided
that illustrate the methodology
The LifeV library: engineering mathematics beyond the proof of concept
LifeV is a library for the finite element (FE) solution of partial
differential equations in one, two, and three dimensions. It is written in C++
and designed to run on diverse parallel architectures, including cloud and high
performance computing facilities. In spite of its academic research nature,
meaning a library for the development and testing of new methods, one
distinguishing feature of LifeV is its use on real world problems and it is
intended to provide a tool for many engineering applications. It has been
actually used in computational hemodynamics, including cardiac mechanics and
fluid-structure interaction problems, in porous media, ice sheets dynamics for
both forward and inverse problems. In this paper we give a short overview of
the features of LifeV and its coding paradigms on simple problems. The main
focus is on the parallel environment which is mainly driven by domain
decomposition methods and based on external libraries such as MPI, the Trilinos
project, HDF5 and ParMetis.
Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar
Weakly Consistent Regularisation Methods for Ill-Posed Problems
This Chapter takes its origin in the lecture notes for a 9 h course at the Institut Henri Poincaré in September 2016. The course was divided in three parts. In the first part, which is not included herein, the aim was to first recall some basic aspects of stabilised finite element methods for convection-diffusion problems. We focus entirely on the second and third parts which were dedicated to ill-posed problems and their approximation using stabilised finite element methods. First we introduce the concept of conditional stability. Then we consider the elliptic Cauchy-problem and a data assimilation problem in a unified setting and show how stabilised finite element methods may be used to derive error estimates that are consistent with the stability properties of the problem and the approximation properties of the finite element space. Finally, we extend the result to a data assimilation problem subject to the heat equation
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
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