22 research outputs found
Further results on a space-time FOSLS formulation of parabolic PDEs
In [2019, Space-time least-squares finite elements for parabolic equations,
arXiv:1911.01942] by F\"uhrer& Karkulik, well-posedness of a space-time
First-Order System Least-Squares formulation of the heat equation was proven.
In the present work, this result is generalized to general second order
parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain
convergence of a standard adaptive finite element method driven by the
least-squares estimator is demonstrated. The proof of the latter easily extends
to a large class of least-squares formulations
Space-time residual minimization for parabolic partial differential equations
Many processes in nature and engineering are governed by partial differential equations (PDEs). We focus on parabolic PDEs, that describe time-dependent phenomena like heat conduction, chemical concentration, and fluid flow. Even if we know that a unique solution exists, we can express it in closed form only under very strict circumstances. So, to understand what it looks like, we turn to numerical approximation. Historically, parabolic PDEs are solved using time-stepping. One first discretizes the PDE in space as to obtain a system of coupled ordinary differential equations in time. This system is then solved using the vast theory for ODEs. While efficient in terms of memory and computational cost, time-stepping schemes take global time steps, which are independent of spatial position. As a result, these methods cannot efficiently resolve details in localized regions of space and time. Moreover, being inherently sequential, they have limited possibilities for parallel computation. In this thesis, we take a different approach and reformulate the parabolic evolution equation as an equation posed in space and time simultaneously. Space-time methods mitigate the aforementioned issues, and moreover produce approximations to the unknown solution that are uniformly quasi-optimal. The focal point of this thesis is the space-time minimal residual (MR) method introduced by R. Andreev, that finds the approximation that minimizes both PDE- and initial error. We discuss its theoretical properties, provide numerical algorithms for its computation, and discuss its applicability in data assimilation (the problem of fusing measured data to its underlying PDE)
Accuracy controlled data assimilation for parabolic problems
This paper is concerned with the recovery of (approximate) solutions to
parabolic problems from incomplete and possibly inconsistent observational
data, given on a time-space cylinder that is a strict subset of the
computational domain under consideration. Unlike previous approaches to this
and related problems our starting point is a regularized least squares
formulation in a continuous infinite-dimensional setting that is based on
stable variational time-space formulations of the parabolic PDE. This allows us
to derive a priori as well as a posteriori error bounds for the recovered
states with respect to a certain reference solution. In these bounds the
regularization parameter is disentangled from the underlying discretization. An
important ingredient for the derivation of a posteriori bounds is the
construction of suitable Fortin operators which allow us to control oscillation
errors stemming from the discretization of dual norms. Moreover, the
variational framework allows us to contrive preconditioners for the discrete
problems whose application can be performed in linear time, and for which the
condition numbers of the preconditioned systems are uniformly proportional to
that of the regularized continuous problem.
In particular, we provide suitable stopping criteria for the iterative
solvers based on the a posteriori error bounds. The presented numerical
experiments quantify the theoretical findings and demonstrate the performance
of the numerical scheme in relation with the underlying discretization and
regularization