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.
In the seminal paper of Bank and Weiser (1985) 17 a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element ...mesh. Despite the promise of Bank–Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel algorithmic approach to constructing hierarchical estimators of the Bank–Weiser type that is designed for implementation in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement for diverse Poisson problems and for mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. Two open source implementations in the DOLFIN and DOLFINx solvers of the FEniCS Project are provided as supplementary material.
•Anisotropic mesh adaptation is applied successfully to turbulent flows on complex geometries, it becomes a reality.•High-fidelity turbulent flow predictions can be obtained with unstructured meshes ...composed only of tetrahedra.•Mesh-converged solutions are achieved in 3D guaranteeing that the computed numerical solution is independent of the mesh.•The mesh adaptation process can correct itself to converge toward the correct solution.
The scope of this paper is to demonstrate the viability of unstructured anisotropic mesh adaptation for commercial aircraft drag and high-lift prediction studies. The main achievement of this work is to demonstrate that mesh-independent certified numerical solutions can be achievable thanks to anisotropic mesh adaptation and that it is possible to run high-fidelity CFD on unstructured adapted meshes composed only of tetrahedra which is fundamental to design robust meshing process for complex geometries. It also points out the early capturing property of the solution-adaptive process in the sense that accurate output functional values are obtained on relatively coarse adapted meshes. On a more practical point of view, this paper demonstrates how mesh adaptation, thanks to its automation, is able to generate meshes that are extremely difficult to envision and almost impossible to generate manually, leading to highly accurate numerical solutions. Moreover, as the process can start from any coarse initial mesh, it greatly simplifies the overall meshing process. This study also analyze the influence of different strategies in the mesh adaptation algorithm and in the error analysis which are key components of the process. Several error estimates are considered: feature-based ones which are based on the standard multiscale Lp interpolation error estimate and goal-oriented ones to control the error on output functionals which rely on an accurate computation of the adjoint state. The adjoint problem proves to be a stiff problem for RANS equations, failing to converge the adjoint state to machine zero may impact negatively the adaptive process. The maturity of the solution-adaptive process is demonstrated on numerous drag and high-lift prediction cases. It has also excelled in sonic boom and turbomachine applications.
•Framework for quantifying the discretization error in soft-tissue simulation.•DWR technique for goal-oriented a posteriori error estimation.•Active properties of the soft-tissue (simplified).•Two ...numerical examples inspired from clinical applications.
Errors in biomechanics simulations arise from modelling and discretization. Modelling errors are due to the choice of the mathematical model whilst discretization errors measure the impact of the choice of the numerical method on the accuracy of the approximated solution to this specific mathematical model. A major source of discretization errors is mesh generation from medical images, that remains one of the major bottlenecks in the development of reliable, accurate, automatic and efficient personalized, clinically-relevant Finite Element (FE) models in biomechanics. The impact of mesh quality and density on the accuracy of the FE solution can be quantified with a posteriori error estimates. Yet, to our knowledge, the relevance of such error estimates for practical biomechanics problems has seldom been addressed, see Bui et al. (2018). In this contribution, we propose an implementation of some a posteriori error estimates to quantify the discretization errors and to optimize the mesh. More precisely, we focus on error estimation for a user-defined quantity of interest with the Dual Weighted Residual (DWR) technique. We test its applicability and relevance in three situations, corresponding to experiments in silicone samples and computations for a tongue and an artery, using a simplified setting, i.e., plane linearized elasticity with contractility of the soft tissue modeled as a pre-stress. Our results demonstrate the feasibility of such methodology to estimate the actual solution errors and to reduce them economically through mesh refinement.
We deal with the goal-oriented error estimates and mesh adaptation for nonlinear partial differential equations. The setting of the adjoint problem and the resulting estimates are not based on a ...differentiation of the primal problem but on a suitable linearization which guarantees the adjoint consistency of the numerical scheme. Furthermore, we develop an efficient adaptive algorithm which balances the errors arising from the discretization and the use of nonlinear as well as linear iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.
We deal with the numerical solution of linear convection–diffusion–reaction equations using the hp-variant of the discontinuous Galerkin method on triangular grids. We develop a mesh adaptive ...algorithm which modifies the size and shape of mesh elements and the corresponding polynomial approximation degrees in order to decrease the error in a target functional under the given tolerance. We recall some theoretical results and describe in detail several technical aspects of this approach. Numerical experiments demonstrating the accuracy, efficiency and robustness of the algorithm are presented.
We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the ...corresponding algebraic systems. The goal-oriented error estimates require the solution of the primal as well as dual algebraic systems. We solve both systems simultaneously using the bi-conjugate gradient method which allows to control the algebraic errors of both systems. We develop a stopping criterion which is cheap to evaluate and guarantees that the estimation of the algebraic error is smaller than the estimation of the discretization error. Using this criterion and an adaptive mesh refinement technique, we obtain an efficient and robust method for the numerical solution of PDEs, which is demonstrated by several numerical experiments.
In this paper, we analyze a multiscale operator splitting method for solving systems of ordinary differential equations such as those that result upon space discretization of a reactiondiffusion ...equation. Our goal is to analyze and accurately estimate the error of the numerical solution, including the effects of any instabilities that can result from multiscale operator splitting. We present both an a priori error analysis and a new type of hybrid a priori-a posteriori error analysis for an operator splitting discontinuous Galerkin finite element method. Both analyses clearly distinguish between the effects of the operator splitting and the discretization of each component of the decomposed problem. The hybrid analysis has the form of a computable a posteriori leading order expression and a provably higher order a priori expression. The hybrid analysis takes into account the fact that the adjoint problems for the original problem and a multiscale operator splitting discretization differ in significant ways. In particular, this provides the means to monitor global instabilities that can arise from operator splitting.
In this paper, we perform an a posteriori error analysis of a multiscale operator decomposition finite element method for the solution of a system of coupled elliptic problems. The goal is to compute ...accurate error estimates that account for the effects arising from multiscale discretization via operator decomposition. Our approach to error estimation is based on a well-known a posteriori analysis involving variational analysis, and the generalized Green's function. Our method utilizes adjoint problems to deal with several new features arising from the multiscale operator decomposition. In part I of this paper, we focus on the propagation of errors arising from the solution of one component to another and the transfer of information between different representations of solution components. We also devise an adaptive discretization strategy based on the error estimates that specifically controls the effects arising from operator decomposition. In part II of this paper, we address issues related to the iterative solution of a fully coupled nonlinear system.
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