Stable multiscale bases and local error estimation for elliptic problems

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.

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

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.

In this work, we consider linear elliptic problems posed in long domains, i.e. the domains whose size in one coordinate direction is much greater than the size in the other directions. If the variation of the coefficients and right‐hand side along the emphasized direction is small, the original problem can be reduced to a lower‐dimensional one that is supposed to be much easier to solve. The a‐posteriori estimation of the error stemming from the model reduction constitutes the goal of the present work. For general coefficient matrix and right‐hand side of the equation, the reliable and efficient error estimator is derived that provides a guaranteed upper bound for the modelling error, exhibits the optimal asymptotics as the size of the domain tends to infinity and correctly indicates the local error distribution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Mesh refinement for anisotropic diffusion in magnetized plasmas

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.

Automatic time step control algorithms for structural dynamics

A simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented which generates C1 surfaces in the limit. The scheme satisfies important requirements for practical applications in computer graphics and engineering. These requirements include the necessity to generate smooth surfaces with local creases and cusps. The scheme can be applied to open nets in which case it generates boundary curves that allow a C0‐join of several subdivision patches. Due to the local support of the scheme, adaptive refinement strategies can be applied. We present a simple device to preserve the consistency of such adaptively refined nets.

Adaptive stochastic Galerkin FEM

In this article we present a complete parallelization approach for simulations of PDEs with applications in optimization and uncertainty quantification. The method of choice for linear or nonlinear elliptic or parabolic problems is the geometric multigrid method since it can achieve optimal (linear) complexity in terms of degrees of freedom, and it can be combined with adaptive refinement strategies in order to find the minimal number of degrees of freedom. This optimal solver is parallelized such that weak and strong scaling is possible for extreme scale HPC architectures. For the space parallelization of the multigrid method we use a tree based approach that allows for an adaptive grid refinement and online load balancing. Parallelization in time is achieved by SDC/ISDC or a space-time formulation. As an example we consider the permeation through human skin which serves as a diffusion model problem where aspects of shape optimization, uncertainty quantification as well as sensitivity to geometry and material parameters are studied. All methods are developed and tested in the UG4 library.

An automatic adaptive refinement procedure for the analysis of shell structures using the nine-node degenerated solid shell element is suggested. The basic adaptive refinement principle and the effects of singularities and boundary layers on the convergence rate of the nine-node element used are discussed. A new stress recovery procedure based on the patch convective co-ordinate system concept is developed for the construction of a continuous smoothed stress field over the shell domains. The stress recovery procedure is easy to implement, requires a modest computational effort and needs only local patch information. It can be applied to shells with non-uniform thickness as well as to multi-layered shell structures. The smoothed recovered stress obtained is then used with the Zienkiewicz and Zhu error estimator for a posteriori error estimation during the adaptive refinement analysis. Numerical results which are in good agreement with theoretical predictions are obtained and they indicate that the current adaptive refinement procedure can eliminate the effect of singularities inside the problem domains so that a near-optimal convergence rate is achieved in all the numerical examples. This also indicates that the stress recovery procedure can produce an accurate stress field and as a result the error estimator can reflect the error distribution of the finite element solution. Even though in the current study only one type of element is used in the analysis, the whole adaptive refinement scheme can be readily applied to any other types of degenerated solid element. © 1997 John Wiley & Sons, Ltd.

Recovery-based error estimation and adaptivity using high-order splines over hierarchical T-meshes

Directional, anisotropic features like layers in the solution of partial differential equations can be resolved favorably by using anisotropic finite element meshes. An adaptive algorithm for such meshes includes the ingredients Error estimation and Information extraction/Mesh refinement. Related articles on a posteriori error estimation on anisotropic meshes revealed that reliable error estimation requires an anisotropic mesh that is aligned with the anisotropic solution. To obtain anisotropic meshes the so‐called Hessian strategy is used, which provides information such as the stretching direction and stretching ratio of the anisotropic elements. This article combines the analysis of anisotropic information extraction/mesh refinement and error estimation (for several estimators). It shows that the Hessian strategy leads to well‐aligned anisotropic meshes and, consequently, reliable error estimation. The underlying heuristic assumptions are given in a stringent yet general form. Numerical examples strengthen the exposition. Hence the analysis provides further insight into a particular aspect of anisotropic error estimation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 625–648, 2002; DOI 10.1002/num.10023

The problem of efficient global error estimation and control is studied in embedded nested implicit Runge–Kutta pairs of Gauss and Lobatto types as applied to stiff ordinary differential equations (ODEs). Stiff problems may arise in many areas of engineering, and their accurate numerical solution is an important issue of computational and applied mathematics. A cheap global error estimation technique designed recently for the mentioned Runge–Kutta pairs can severely overestimate the global error when applied to stiff ODEs and, hence, this reduces the efficiency of those solvers. In the present paper, we explain the cause of that error overestimation and show how to improve the mentioned computation techniques for stiff problems. Such modifications not only boost the efficiency of numerical integration of stiff ODEs, but also make the embedded nested implicit Runge–Kutta pairs with scaled modified local and global error controls superior to stiff built-in MATLAB ODE solvers with only local error control when applied to important test examples.

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