587 research outputs found
Linearization Errors in Discrete Goal-Oriented Error Estimation
Goal-oriented error estimation provides the ability to approximate the
discretization error in a chosen functional quantity of interest. Adaptive mesh
methods provide the ability to control this discretization error to obtain
accurate quantity of interest approximations while still remaining
computationally feasible. Traditional discrete goal-oriented error estimates
incur linearization errors in their derivation. In this paper, we investigate
the role of linearization errors in adaptive goal-oriented error simulations.
In particular, we develop a novel two-level goal-oriented error estimate that
is free of linearization errors. Additionally, we highlight how linearization
errors can facilitate the verification of the adjoint solution used in
goal-oriented error estimation. We then verify the newly proposed error
estimate by applying it to a model nonlinear problem for several quantities of
interest and further highlight its asymptotic effectiveness as mesh sizes are
reduced. In an adaptive mesh context, we then compare the newly proposed
estimate to a more traditional two-level goal-oriented error estimate. We
highlight that accounting for linearization errors in the error estimate can
improve its effectiveness in certain situations and demonstrate that localizing
linearization errors can lead to more optimal adapted meshes
Goal-oriented error estimation for transient parabolic problems
This work focuses on controlling the error and adapting the discretization in the context of parabolic problems. In order to obtain a sound mathematical framework, the time domain is discretized using a Discontinuous Galerkin (DG) approach. This
allows to formulate the time stepping procedure in a variational format. The error is measured in the basis of an output of interest of the solution, defined by a linear functional. A dual problem, associated with this linear output is introduced.
The dual problem has to be solved backward in time.
An error representation is introduced, based on the weak residual of the primal error applied to the dual solution. Two different alternatives are studied to estimate the error in the dual solution: 1) recovery based error estimators and 2) implicit
residual type estimators. Once the error assessment is performed implicitly in the dual problem, the obtained estimate is plugged into the primal residual to obtain the error in the quantity of interest. The implementation of the estimator
is drastically simplified by using the weak version of the residual instead of the strong version used in previous works.
Thus, the output error is assessed using a mixed technique, explicit for the primal problem and implicit for the dual. In the framework of adaptive computations of transient problems, this approach is very attractive because it allows using first
the implicit scheme for the dual problem and then integrating the primal problem, estimating the error explicitly and eventually adapting the space-time grid. Thus, at every time step of the time marching scheme, the estimate of the dual error is injected into the primal residual (explicit estimate for the primal problem)
Goal-oriented error estimation for fluid-structure interaction problems
In this work, we present an adaptive finite element method for the numerical simulation of stationary fluid-structure interaction problems. The coupled system is given in a variational and monolithic Arbitrary Lagrangian Eulerian framework. We derive methods for goal-oriented error estimation and mesh adaptation with the dual weighted residual method. Key to applying this error estimator is the underlying canonic variational formulation of the fluid-structure interaction problem by mapping the flow problem to ALE coordinates. The developed method is applied to two and three dimensional stationary benchmark problems coupling the incompressible Navier-Stokes equations with a nonlinear hyper-elastic material law
Goal oriented error estimation for the element free Galerkin method
A novel approach for implicit residual-type error estimation in meshfree methods is presented. This allows to compute upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh-free methods
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