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 and its implementation in an advanced software architecture. The error estimate allows the separation of the temporal and spatial discretization error contributions which facilitates the simultaneous adjustment of the time and space mesh. The performance of the approach and its software implementation is studied by numerical convergence examples as well as an example of physical interest for convection-dominated cases.
We present an effective adaptive procedure for the numerical approximation of the steady-state Gross-Pitaevskii equation. Our approach is solely based on energy minimization, and consists of a combination of gradient flow iterations and adaptive finite element mesh refinements. Numerical tests show that this strategy is able to provide highly accurate results, with optimal convergence rates with respect to the number of freedom.
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 …
We consider the Virtual Element method (VEM) introduced by Beirão da Veiga, Lovadina and Vacca in 2016 for the numerical solution of the steady, incompressible Navier-Stokes equations; the method has arbitrary order $k \geq 2$ and guarantees divergence-free velocities. For such discretization, we develop a residual-based a posteriori error estimator, which is a combination of standard terms in VEM analysis (residual terms, data oscillation, and VEM stabilization), plus some other terms originated by the VEM discretization of the nonlinear convective term. We show that a linear combination of t…
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）
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-intrusi…
We consider finite element discretizations of Maxwell's equations coupled with a non-local hydrodynamic Drude model that accurately accounts for electron motions in metallic nanostructures. Specifically, we focus on a posteriori error estimation and mesh adaptivity, which is of particular interest since the electromagnetic field usually exhibits strongly localized features near the interface between metals and their surrounding media. We propose a novel residual-based error estimator that is shown to be reliable and efficient. We also present a set of numerical examples where the estimator dri…
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…
High-frequency wave propagation has many important applications in acoustics, elastodynamics, and electromagnetics. Unfortunately, the finite element discretization for these problems suffers from significant numerical pollution errors that increase with the wavenumber. It is critical to control these errors to obtain a stable and accurate method. We study the effect of pollution for very long waveguide problems in the context of robust discontinuous Petrov-Galerkin (DPG) finite element discretizations. Our numerical experiments show that the pollution primarily has a diffusive effect causing …
Volume Term Adaptivity for Discontinuous Galerkin Schemes
RWTH Aachen University（亚琛工业大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；Johannes Gutenberg University Mainz（约翰内斯·古腾堡美因茨大学）；University of Augsburg（奥格斯堡大学）；University of Cologne（科隆大学）
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 consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a suitable linearization which guarantees the adjoint consistency of the numerical scheme. We derive error estimates and develop an efficient adaptive algorithm which balances the errors arising from the discretization and use of iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exp…
Variational formulations of time-dependent PDEs in space and time yield $(d+1)$-dimensional problems to be solved numerically. This increases the number of unknowns as well as the storage amount. On the other hand, this approach enables adaptivity in space and time as well as model reduction w.r.t. both type of variables. In this paper, we show that matrix oriented techniques can significantly reduce the computational timings for solving the arising linear systems outperforming both time-stepping schemes and other solvers.
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…
Convergence failure and slow convergence rates are among the biggest challenges with solving the system of non-linear equations numerically. Although mitigated, such issues still linger when using strictly small time steps and unconditionally stable fully implicit schemes. The price that comes with restricting time steps to small scales is the enormous computational load, especially in large-scale models. To address this problem, we introduce a sequential local mesh refinement framework to optimize convergence rate and prevent convergence failure, while not restricting the whole system to smal…
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…
In this paper, we present a predictor-corrector strategy for constructing rank-adaptive dynamical low-rank approximations (DLRAs) of matrix-valued ODE systems. The strategy is a compromise between (i) low-rank step-truncation approaches that alternately evolve and compress solutions and (ii) strict DLRA approaches that augment the low-rank manifold using subspaces generated locally in time by the DLRA integrator. The strategy is based on an analysis of the error between a forward temporal update into the ambient full-rank space, which is typically computed in a step-truncation approach before …
We present an accurate and efficient discretization approach for the adaptive discretization of typical model equations employed in numerical weather prediction. A semi-Lagrangian approach is combined with the TR-BDF2 semi-implicit time discretization method and with a spatial discretization based on adaptive discontinuous finite elements. The resulting method has full second order accuracy in time and can employ polynomial bases of arbitrarily high degree in space, is unconditionally stable and can effectively adapt the number of degrees of freedom employed in each element, in order to balanc…
