Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to π 2 ${\frac{\pi }{2}}$ . The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allow the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks.

Model order reduction (MOR) is usually equipped with two stages: the offline stage and the online stage.For many intrusive MOR methods that are based on projection, the offline stage is usually realized via a greedy algorithm.The greedy algorithm is used to properly select important parameter samples that contribute most to the solution space.The offline computational time is basically the runtime of the greedy algorithm.For large-scale systems, the offline computation is expensive and the runtime is longer than several hours even if it is run on a high-performance server.Sometimes, the system is not very large, for example, the number of degrees of freedom is only O(10 5 ), but the system structure is complicated such that the greedy algorithm still takes long time to converge.Once the reduced-order model is obtained, the online stage of querying the reduced-order model for simulation is very fast and in real-time.This work concerns a strategy of significantly accelerating the greedy algorithm.We propose multifidelity error estimators and replace the high-fidelity error estimator in the greedy algorithm.In fact, a bi-fidelity error estimation has been used in the adaptive greedy algorithm proposed in [1] without being formally defined.To further improve the convergence speed of the greedy algorithm, we propose multi-fidelity error estimation built upon the bi-fidelity error estimation.Here, we use a more efficient high-fidelity error estimator than the two different high-fidelity error estimators used in [1].Although the proposed multi-fidelity error estimation is dependent on the original high-fidelity error estimator, the concept of using multi-fidelity error estimation is generic and can be extended to develop multi-fidelity error estimation associated with other high-fidelity error estimators.The multi-fidelity error estimator is then used to adaptively update the training set of the greedy algorithm, keeping the offline computational costs on a possibly lowest level.Consequently, the computational complexity at each iteration of the greedy algorithm is reduced and the algorithm converges more than 3 times faster without incurring noticeable accuracy loss.

19 - Viscous and Non-Newtonian Flows

Comparison of Eulerian and Lagrangian numerical techniques for the Stokes equations in the presence of strongly varying viscosity

The problem of creeping flow past a swarm of porous circular cylinders with Happel and Kuwabara boundary conditions is investigated. The Brinkman equation for the flow inside the porous cylinder and the Stokes equation outside the porous cylinder in their stream function formulations are used. The force experienced by each porous circular cylinder in a cell is evaluated. Explicit expressions of stream functions are obtained for both the inside and outside flow fields. The earlier results reported by Happel and Kuwabara for flow past a solid cylinder in Happel’s and Kuwabara’s cell model, have been deduced. Analytical expressions for the velocity components, pressure, vorticity and stresstensor are also obtained

This paper concerns the Stokes flow of an incompressible viscous fluid past a swarm of porous nanocylindrical particles enclosing a solid cylindrical core with Kuwabara boundary condition. An aggregate of porous nanocylindrical particles is considered as a hydro‐dynamically equivalent to a solid cylindrical core with concentric porous cylindrical shell. The Brinkman equation inside the porous cylindrical shell and the Stokes equation outside the porous cylindrical shell in their stream function formulations are used. Explicit expressions for the stream functions in both regions have been investigated. The drag force acting at each nanoporous cylindrical particle in a cell is evaluated. Also, we solved the same problem by using Happel boundary condition on the hypothetical cell. In certain limiting cases, drag force converges to pre‐existing analytical results, such as the drag on a porous circular cylinder and the drag on a solid cylinder in Kuwabara′s cell or Happel′s cell. Representative results are then discussed and compared for both cases and presented in graphical form by using Mathematica software.

We consider a surface Stokes problem in stream function formulation on a simply connected oriented surface Γ ⊂ ℝ3 without boundary. This formulation leads to a coupled system of two second order scalar surface partial differential equations (for the stream function and an auxiliary variable). To this coupled system a trace finite element discretization method is applied. The main topic of the paper is an error analysis of this discretization method, resulting in optimal order discretization error bounds. The analysis applies to the surface finite element method of Dziuk–Elliott, too. We also investigate methods for reconstructing velocity and pressure from the stream function approximation. Results of numerical experiments are included.

A reliable, efficient and localized error estimator for a discontinuous Galerkin method for the Signorini problem

Drag force on a circular cylinder midway between two parallel plates at [formula omitted] Part 2: moving uniformly (numerical and experimental)

Drag force on a circular cylinder midway between two parallel plates at very low Reynolds numbers—Part 1: Poiseuille flow (numerical)

We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. We prove optimal a priori estimates for both stream function and vorticity, and present numerical results to demonstrate the efficiency of the approach.

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three‐dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable , we rewrite the problem into a two‐order system. Then, the a priori error estimates are derived in L2 norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r () are used, we obtain the optimal convergence rate of order r + 1 in L2 norm and of order r in DG norm for u, and the order r in both norms for . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017

In this article, we propose a residual based reliable and efficient error estimator for the new dual mixed finite element method of the elasticity problem in a polygonal domain, introduced by M. Farhloul and M. Fortin. With the help of a specific generalized Helmholtz decomposition of the error on the strain tensor and the classical decomposition of the error on the gradient of the displacements, we show that our global error estimator is reliable. Efficiency of our estimator follows by using classical inverse estimates. The lower and upper error bounds obtained are uniform with respect to the Lamé coefficient λ, in particular avoiding locking phenomena. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.

Aerosol coagulation is a significant process regarding the dynamics of aerosols in the atmosphere. This process leads to an evolution of the size distribution of aerosols over time induced by particle collisions, and is described by Smoluchowski equation [1].Accurate numerical simulations of this process are computationally demanding by typical discretization methods [2]. We investigate the effectiveness of the reduced basis method [3] to provide accurate but less demanding simulations, by constructing a reduced order subspace included within the reference high-dimensional space used to provide high fidelity simulations.We also provide residual-based a posteriori error estimates [4] which enable certification of results up to a given error tolerance. We obtain efficient online model and error estimates as their online computational cost only scale with the reduced dimension, and not the dimension of the high-fidelity model.By careful design of reduced subspaces we ensure that some properties of the high fidelity operator, such as mass conservation [5], are also preserved by reduced order models.[1] M. V. Smoluchowski. Drei vortage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen. Physik, 17, 557–585, 1916[2] E. Debry, and B. Sportisse. Solving aerosol coagulation with size-binning methods. Applied Numerical Mathematics, 57(9), 1008–1020, 2007[3] A. Quarteroni, et al. Reduced Basis Methods for Partial Differential Equations. Springer Cham. 2015[4] M. Grepl, and A. Patera, A Posteriori Error Bounds for Reduced-Basis Approximations of Parametrized Parabolic Partial Differential EquationsESAIM: Mathematical Modelling and Numerical Analysis, 39(1), 157-181, 2005[5] F. Filbet, and P. Laurençot. Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation. Archiv der Mathematik, 83(6), 558-567, 2004

We present an adaptive refinement strategy for the h-version of the boundary element method with weakly singular operators on surfaces. The model problem deals with the exterior Stokes problem, and thus considers vector functions. Our error indicators are computed by local projections onto ID subspaces defined by mesh refinement. These indicators measure the error separately for the vector components and allow for component-independent adaption. Assuming a saturation condition, the indicators give rise to an efficient and reliable error estimator. Also we describe how to deal with meshes containing quadrilaterals which are not shape regular. The theoretical results are underlined by numerical experiments. To justify the saturation assumption, in the Appendix we prove optimal lower a priori error estimates for edge singularities on uniform and graded meshes.