22 research outputs found
Forward-in-Time Goal-Oriented Adaptivity
In goal-oriented adaptive algorithms for partial differential equations, we adapt the finite element mesh in order to reduce the error of the solution in some quantity of interest. In time-dependent problems, this adaptive algorithm involves solving a dual problem that runs backward in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we define a pseudo-dual problem that runs forward in time. We also describe a forward-in-time adaptive algorithm that works for some specific problems. Although it is not possible to define a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one-dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. Finally, we propose a hybrid algorithm that employs the classical backward-in-time dual problem once and then performs the adaptive process forwards in time
Pointwise best approximation results for Galerkin finite element solutions of parabolic problems
In this paper we establish a best approximation property of fully discrete
Galerkin finite element solutions of second order parabolic problems on convex
polygonal and polyhedral domains in the norm. The discretization
method uses of continuous Lagrange finite elements in space and discontinuous
Galerkin methods in time of an arbitrary order. The method of proof differs
from the established fully discrete error estimate techniques and for the first
time allows to obtain such results in three space dimensions. It uses elliptic
results, discrete resolvent estimates in weighted norms, and the discrete
maximal parabolic regularity for discontinuous Galerkin methods established by
the authors in [16]. In addition, the proof does not require any relationship
between spatial mesh sizes and time steps. We also establish a local best
approximation property that shows a more local behavior of the error at a given
point
Unsteady Output-Based Adaptation Using Continuous-in-Time Adjoints
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143081/1/6.2017-0529.pd
On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations
The aim of this paper is to extend the global error estimation and control
addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value
problems to finite difference solutions of semilinear parabolic partial
differential equations. The approach presented there is combined with an
estimation of the PDE spatial truncation error by Richardson extrapolation to
estimate the overall error in the computed solution. Approximations of the
error transport equations for spatial and temporal global errors are derived by
using asymptotic estimates that neglect higher order error terms for
sufficiently small step sizes in space and time. Asymptotic control in a
discrete -norm is achieved through tolerance proportionality and uniform
or adaptive mesh refinement. Numerical examples are used to illustrate the
reliability of the estimation and control strategies
Goal-Oriented Adaptivity in Space-Time Finite Element Simulations of Nonstationary Incompressible Flows
Subject of this paper is the development of an a posteriori error estimator for nonstationary incompressible flow problems. The error estimator is computable and able to assess the temporal and spatial discretization errors separately. Thereby, the error is measured in an arbitrary quantity of interest because measuring errors in global norms is often of minor importance in practical applications. The basis for this is a finite element discretization in time and space. The techniques presented here also provide local error indicators which are used to adaptively refine the temporal and spatial discretization. A key ingredient in setting up an efficient discretization method is balancing the error contributions due to temporal and spatial discretization. To this end, a quantitative assessment of the individual discretization errors is required. The described method is validated by an established Navier-Stokes benchmark
Variational Formulations for Explicit Runge-Kutta Methods
Variational space-time formulations for partial di fferential equations have been of great interest in the last decades, among other things, because they allow to develop mesh-adaptive algorithms. Since it is known that
implicit time marching schemes have variational structure, they are often employed for adaptivity. Previously, Galerkin formulations of explicit methods were introduced for ordinary di fferential equations employing speci fic
inexact quadrature rules. In this work, we prove that the explicit Runge-Kutta methods can be expressed as discontinuous-in-time Petrov-Galerkin methods for the linear di ffusion equation. We systematically build trial
and test functions that, after exact integration in time, lead to one, two, and general stage explicit Runge-Kutta methods. This approach enables us to reproduce the existing time-domain (goal-oriented) adaptive algorithms
using explicit methods in time
A goal oriented error estimator and mesh adaptivity for sea ice simulations
For the first time we introduce an error estimator for the numerical
approximation of the equations describing the dynamics of sea ice. The idea of
the estimator is to identify different error contributions coming from spatial
and temporal discretization as well as from the splitting in time of the ice
momentum equations from further parts of the coupled system. The novelty of the
error estimator lies in the consideration of the splitting error, which turns
out to be dominant with increasing mesh resolution. Errors are measured in user
specified functional outputs like the total sea ice extent. The error estimator
is based on the dual weighted residual method that asks for the solution of an
additional dual problem for obtaining sensitivity information. Estimated errors
can be used to validate the accuracy of the solution and, more relevant, to
reduce the discretization error by guiding an adaptive algorithm that optimally
balances the mesh size and the time step size to increase the efficiency of the
simulation