We introduce a goal-oriented strategy for multiscale computations performed using the Multiscale Finite Element Method (MsFEM). In a previous work, we have shown how to use, in the MsFEM framework, the concept of Constitutive Relation Error (CRE) to obtain a guaranteed and fully computable a posteriori error estimate in the energy norm (as well as error indicators on various error sources). Here, the CRE concept is coupled with the solution of an adjoint problem to control the error and drive an adaptive procedure with respect to a given output of interest. Furthermore, a local and non-intrusive enrichment technique is proposed to enhance the accuracy of error bounds. The overall strategy, which is fully automatic and robust, enables to reach an appropriate trade-off between certified reliability and computational cost in the MsFEM context. The performances of the proposed method are investigated on several illustrative numerical test cases. In particular, the error estimation is observed to be very accurate, yielding a very efficient adaptive procedure.
We consider an adaptive finite element method with arbitrary but fixed polynomial degree $p \ge 1$, where adaptivity is driven by an edge-based residual error estimator. Based on the modified maximum criterion from [Diening et al, Found. Comput. Math. 16, 2016], we propose a goal-oriented adaptive algorithm and prove that it is instance optimal. More precisely, the goal-error is bounded by the product of the total errors (being the sum of energy error plus data oscillations) of the primal and the dual problem, and the proposed algorithm is instance optimal with respect to this upper bound. Num…
We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint …
In this work, we consider multigoal-oriented error estimation for stationary fluid-structure interaction. The problem is formulated within a variational-monolithic setting using arbitrary Lagrangian-Eulerian coordinates. Employing the dual-weighted residual method for goal-oriented a posteriori error estimation, adjoint sensitivities are required. For multigoal-oriented error estimation, a combined functional is formulated such that several quantities of interest are controlled simultaneously. As localization technique for mesh refinement we employ a partition-of-unity. Our algorithmic develop…
Mathematical modeling and numerical multigoal-oriented a posteriori error control and adaptivity for a stationary, nonlinear, coupled flow temperature model with temperature dependent density
Goal-oriented space-time adaptivity for the Navier--Stokes equations based on the dual weighted residual method
Helmut Schmidt University Hamburg, Faculty of Mechanical and Civil Engineering（汉堡赫尔穆特·施密特大学机械与土木工程学院）；University of Magdeburg, Institute for Analysis and Numerics（马格德堡大学分析与数值研究所）
In this work, a flexible higher-order space-time adaptive finite element approximation of convection-dominated transport with coupled fluid flow is developed and studied. Convection-dominated transport is a challenging subproblem in poromechanics in which coupled transport with flow, chemical reaction and mechanical response in porous media is considered. Key ingredients are the arbitrary degree discontinuous Galerkin time discretization of the primal and dual problems for the Dual Weighted Residual (DWR) approach, an a posteriori error estimation for the transport problem coupled with flow an…
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 cost- and memory-efficient numerical simulation of coupled volume-based multi-physics problems like flow, transport, wave propagation and others remains a challenging task with finite element method (FEM) approaches. Goal-oriented space and time adaptive methods derived from the dual weighted residual (DWR) method appear to be a shiny key technology to generate optimal space-time meshes to minimise costs. Current implementations for challenging problems of numerical screening tools including the DWR technology broadly suffer in their extensibility to other problems, in high memory consumpt…
In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier-Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor-Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition of unity based error localization. We derive th…
During the last decade there has been an increase on the use of goal-oriented error estimates which help to quantify and control the local error on a quantity of interest (QoI) that might result relevant for design purposes (e.g. the mean stress or mean displacement in a particular area, the stress intensity factor for fracture problems,...). Residual-based error estimators have been used to estimate the error in quantities of interest for finite element approximations. This work presents a recovery-based error estimation technique for QoI whose main characteristic is the use of an enhanced ve…
Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation
TOBB University of Economics and Technology（TOBB经济与技术大学）；Louisiana State University（路易斯安那州立大学）；Sandia National Laboratories（桑迪亚国家实验室）；University of California San Diego（加州大学圣地亚哥分校）
We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous Galerkin (DG) method with focus on the goal-oriented error estimates and adaptivity. We analyze the adjoint consistency of the DG scheme where the dual problem is not formulated by the differentiation of the DG form and the target functional but using a suitable linearization of the nonlinear forms. Further, we present the goal-oriented anisotropic $hp$-mesh adaptation technique for the Euler equations. The theoretical results are supported by numerical experiments.
In this work, we implement goal-oriented error control and spatial mesh adaptivity for stationary fluid-structure interaction. The a posteriori error estimator is realized using the dual-weighted residual method in which the adjoint equation arises. The fluid-structure interaction problem is formulated within a variational-monolithic framework using arbitrary Lagrangian-Eulerian coordinates. The overall problem is nonlinear and solved with Newton's method. We specifically consider the FSI-1 benchmark problem in which quantities of interest include the elastic beam displacements, drag, and lift…
We show how a posteriori goal oriented error estimation can be used to efficiently solve the subproblems occurring in a Model Predictive Control (MPC) algorithm. In MPC, only an initial part of a computed solution is implemented as a feedback, which motivates grid refinement particularly tailored to this context. To this end, we present a truncated cost functional as objective for goal oriented adaptivity and prove under stabilizability assumptions that error indicators decay exponentially outside the support of this quantity. This leads to very efficient time and space discretizations for MPC…
Multigoal-oriented optimal control problems with nonlinear PDE constraints
Johannes Kepler University（约翰·开普勒大学）；Austrian Academy of Sciences（奥地利科学院）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；Technische Universität Darmstadt（达姆施塔特工业大学）
In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint problem is solved in strong form with a feedforward neural network using two or three hidden layers. The main objective of our approach is to explore alternatives for solving the adjoint problem with greater potential of a numerical cost reduction. The proposed algorithm is based on the general goal-oriented error estimation theorem including both linear an…
