Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

Error analysis of the Galerkin FEM in L 2 -based norms for problems with layers: On the importance, conception and realization of balancing

In the present thesis it is shown that the most natural choice for a norm for the analysis of the Galerkin FEM, namely the energy norm, fails to capture the boundary layer functions arising in certain reaction-diffusion problems. In view of a formal Definition such reaction-diffusion problems are not singularly perturbed with respect to the energy norm. This observation raises two questions: 1. Does the Galerkin finite element method on standard meshes yield satisfactory approximations for the reaction-diffusion problem with respect to the energy norm? 2. Is it possible to strengthen the energy norm in such a way that the boundary layers are captured and that it can be reconciled with a robust finite element method, i.e.~robust with respect to this strong norm? In Chapter 2 we answer the first question. We show that the Galerkin finite element approximation converges uniformly in the energy norm to the solution of the reaction-diffusion problem on standard shape regular meshes. These results are completely new in two dimensions and are confirmed by numerical experiments. We also study certain convection-diffusion problems with characterisitc layers in which some layers are not well represented in the energy norm. These theoretical findings, validated by numerical experiments, have interesting implications for adaptive methods. Moreover, they lead to a re-evaluation of other results and methods in the literature. In 2011 Lin and Stynes were the first to devise a method for a reaction-diffusion problem posed in the unit square allowing for uniform a priori error estimates in an adequate so-called balanced norm. Thus, the aforementioned second question is answered in the affirmative. Obtaining a non-standard weak formulation by testing also with derivatives of the test function is the key idea which is related to the H^1-Galerkin methods developed in the early 70s. Unfortunately, this direct approach requires excessive smoothness of the finite element space considered. Lin and Stynes circumvent this problem by rewriting their problem into a first order system and applying a mixed method. Now the norm captures the layers. Therefore, they need to be resolved by some layer-adapted mesh. Lin and Stynes obtain optimal error estimates with respect to the balanced norm on Shishkin meshes. However, their method is unable to preserve the symmetry of the problem and they rely on the Raviart-Thomas element for H^div-conformity. In Chapter 4 of the thesis a new continuous interior penalty (CIP) method is present, embracing the approach of Lin and Stynes in the context of a broken Sobolev space. The resulting method induces a balanced norm in which uniform error estimates are proven. In contrast to the mixed method the CIP method uses standard Q_2-elements on the Shishkin meshes. Both methods feature improved stability properties in comparison with the Galerkin FEM. Nevertheless, the latter also yields approximations which can be shown to converge to the true solution in a balanced norm uniformly with respect to diffusion parameter. Again, numerical experiments are conducted that agree with the theoretical findings. In every finite element analysis the approximation error comes into play, eventually. If one seeks to prove any of the results mentioned on an anisotropic family of Shishkin meshes, one will need to take advantage of the different element sizes close to the boundary. While these are ideally suited to reflect the solution behavior, the error analysis is more involved and depends on anisotropic interpolation error estimates. In Chapter 3 the beautiful theory of Apel and Dobrowolski is extended in order to obtain anisotropic interpolation error estimates for macro-element interpolation. This also sheds light on fundamental construction principles for such operators. The thesis introduces a non-standard finite element space that consists of biquadratic C^1-finite elements on macro-elements over tensor product grids, which can be viewed as a rectangular version of the C^1-Powell-Sabin element. As an application of the general theory developed, several interpolation operators mapping into this FE space are analyzed. The insight gained can also be used to prove anisotropic error estimates for the interpolation operator induced by the well-known C^1-Bogner-Fox-Schmidt element. A special modification of Scott-Zhang type and a certain anisotropic interpolation operator are also discussed in detail. The results of this chapter are used to approximate the solution to a recation-diffusion-problem on a Shishkin mesh that features highly anisotropic elements. The obtained approximation features continuous normal derivatives across certain edges of the mesh, enabling the analysis of the aforementioned CIP method.:Notation 1 Introduction 2 Galerkin FEM error estimation in weak norms 2.1 Reaction-diffusion problems 2.2 A convection-diffusion problem with weak characteristic layers and a Neumann outflow condition 2.3 A mesh that resolves only part of the exponential layer and neglects the weaker characteristic layers 2.3.1 Weakly imposed characteristic boundary conditions 2.4 Numerical experiments 2.4.1 A reaction-diffusion problem with boundary layers 2.4.2 A reaction-diffusion problem with an interior layer 2.4.3 A convection-diffusion problem with characteristic layers and a Neumann outflow condition 2.4.4 A mesh that resolves only part of the exponential layer and neglects the weaker characteristic layers 3 Macro-interpolation on tensor product meshes 3.1 Introduction 3.2 Univariate C1-P2 macro-element interpolation 3.3 C1-Q2 macro-element interpolation on tensor product meshes 3.4 A theory on anisotropic macro-element interpolation 3.5 C1 macro-interpolation on anisotropic tensor product meshes 3.5.1 A reduced macro-element interpolation operator 3.5.2 The full C1-Q2 interpolation operator 3.5.3 A C1-Q2 macro-element quasi-interpolation operator of Scott-Zhang type on tensor product meshes 3.5.4 Summary: anisotropic C1 (quasi-)interpolation error estimates 3.6 An anisotropic macro-element of tensor product type 3.7 Application of macro-element interpolation on a tensor product Shishkin mesh 4 Balanced norm results for reaction-diffusion 4.1 The balanced finite element method of Lin and Stynes 4.2 A C0 interior penalty method 4.3 Galerkin finite element method 4.3.1 L2-norm error bounds and supercloseness 4.3.2 Maximum-norm error bounds 4.4 Numerical verification 4.5 Further developments and summary Reference

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Convergence analysis of a weak Galerkin finite element method on a Shishkin mesh for a singularly perturbed fourth-order problem in 2D

We consider the singularly perturbed fourth-order boundary value problem $\varepsilon ^{2}\Delta ^{2}u-\Delta u=f$ on the unit square $\Omega \subset \mathbb{R}^2$, with boundary conditions $u = \partial u / \partial n = 0$ on $\partial \Omega$, where $\varepsilon \in (0, 1)$ is a small parameter. The problem is solved numerically by means of a weak Galerkin(WG) finite element method, which is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on finite element partitions consisting of polygons of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with $N^2$ elements is constructed ,convergence of the method is proved in a discrete $H^2$ norm for the corresponding WG finite element solutions and numerical results are presented

The Investigation of Efficiency of Physical Phenomena Modelling Using Differential Equations on Distributed Systems

This work is dedicated to development of mathematical modelling software. In this dissertation numerical methods and algorithms are investigated in software making context. While applying a numerical method it is important to take into account the limited computer resources, the architecture of these resources and how do methods affect software robustness. Three main aspects of this investigation are that software implementation must be efficient, robust and be able to utilize specific hardware resources. The hardware specificity in this work is related to distributed computations of different types: single CPU with multiple cores, multiple CPUs with multiple cores and highly parallel multithreaded GPU device. The investigation is done in three directions: GPU usage for 3D FDTD calculations, FVM method usage to implement efficient calculations of a very specific heat transferring problem, and development of special techniques for software for specific bacteria self organization problem when the results are sensitive to numerical methods, initial data and even computer round-off errors. All these directions are dedicated to create correct technological components that make a software implementation robust and efficient. The time prediction model for 3D FDTD calculations is proposed, which lets to evaluate the efficiency of different GPUs. A reasonable speedup with GPU comparing to CPU is obtained. For FVM implementation the OpenFOAM open source software is selected as a basis for implementation of calculations and a few algorithms and their modifications to solve efficiency issues are proposed. The FVM parallel solver is implemented and analyzed, it is adapted to heterogeneous cluster Vilkas. To create robust software for simulation of bacteria self organization mathematically robust methods are applied and results are analyzed, the algorithm is modified for parallel computations

Singularly perturbed boundary value problems with two parameters on various meshes

U tezi se istražuje uniformna konvergencija Galerkinovog postupka konačnih elemenata na mrežama različitog tipa za dvoparametarske singularno perturbovane probleme. Uvedene su slojno-adaptivne mreže za probleme konvekcije-reakcije-difuzije:  Bahvalovljeva, Duran-Šiškinova i Duranova za jednodimenzionalni i Duran-Šiškinova i Duranova mreža za dvodimenzionalni problem. Za pomenute probleme na svim ovim mrežama analizirane su greške interpolacije, diskretizacije i greška u energetskoj normi i dokazana je uniformna konvergencija Galerkinovog postupka konačnih elemenata. Sva teorijska tvrđenja su potvrđena numeričkim eksperimentima.  The thesis explores the uniform convergence for Galerkin nite element method on various meshes for two parameter singularly perturbed problems. Layer-adapted meshes are introduced for convection-reaction-diusion problems: Bakhvalov, Duran-Shishkin and Duran meshes for a one dimensional and Duran-Shishkin and Duran meshes for a two dimensional problem. We analyze the errors of interpolation, discretization and error in the energy norm and prove the parameter uniform convergence for Galerkin nite element method on mentioned meshes. Numerical experiments support theoretical ndings.

Robust arbitrary order mixed finite element methods for the incompressible Stokes equations

Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H1-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H1 and L2 errors of the velocity and the L2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results

Robust arbitrary order mixed finite element methods for the incompressible Stokes equations

Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H1-conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H1 and L2 errors of the velocity and the L2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results

Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems