Averaging is an important method to extract effective macroscopic dynamics from complex systems w... more Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Importantly, the deviation between the original equation and the averaged equation is also studied. A martingale approach proves that the deviation is described by a Gaussian process. This gives an approximation to errors of O(ǫ) instead of O( √ ǫ) attained in previous averaging.
We present a self-paced, instructor-assisted approach to teaching Linear Algebra, as an alternati... more We present a self-paced, instructor-assisted approach to teaching Linear Algebra, as an alternative to traditional classroom instruction. In the self-paced model students learn by doing, at their own pace, instead of listening to an instructor and taking notes. The instructor is there to motivate them,
Many problems in computational engineering and science, such as solid and fluid mechanics, electr... more Many problems in computational engineering and science, such as solid and fluid mechanics, electromagnetics, heat transfer, or chemistry, are sufficiently well described on the macroscopic level in terms of partial differential equations (PDEs). In practice, these processes may be very complex, and the presence of multiple spatial and/or temporal scales, or even discontinuities in the solution, often makes their computer simulation challenging. There exist advanced numerical methods to tackle these problems, such as finite element methods (FEM). Lately, new advanced version of these methods have appeared, such as hierarchic higher-order finite element methods (hp-FEM) and extended finite element methods (X-FEM). Most of these methods work on a traditional basis where no uncertainty considerations are present in the modeling or computation. However, the need for numerical treatment of uncertainty becomes increasingly urgent. In many cases a given problem can be solved efficiently and accurately for a given set of input data (such as geometry, boundary conditions, material parameters, etc.), but little can be said about how the solution depends on uncertainties in these parameters. However, the design of an engineered system requires the performance of the system to be guaranteed over its lifetime. One of the major difficulties a designer must face is that neither the external demands of the systems nor its manufacturing variations are known exactly. In order to overcome this uncertainty, the designer must provide excessive capabilities and over design the system. As analysis tools continue to be developed, the predictive skills of designers have become finer. In addition, the demands of the market place require that more efficient designs be developed. In order to satisfy these current requirements in designs subject to uncertainties, the uncertainties in the performance of the system must be included in the analysis. At present, analytical and Monte-Carlo techniques are used to handle probabilistic uncertainty, and interval finite element methods are used to handle interval uncertainty. In many practical situations, we have both probabilistic and interval uncertainty. The problem of efficient combination of
This chapter is devoted to the discussion of selected topics related to finite element software. ... more This chapter is devoted to the discussion of selected topics related to finite element software. In Section B. 1 we present an efficient way of connecting the packages PETSc, Trilinos and UMFPACK to a finite element solver. Section B.2 gives a brief description of the highperformance modular finite element system HERMES. The full manual posted on our web page offers additional technical details. At the end of Section B.2 we present numerical results obtained with HERMES, where the efficiency of the lowest-order FEM and the hp-FEM is compared. Since it was not possible to include color pictures with this book, a color PDF file with the visualizations is available on our web page. Efficient solvers for sparse systems of linear algebraic equations are key ingredients of finite element programs. Nowadays an engineer or researcher hardly can afford developing matrix solvers on hisher own, and thus public domain software packages play an increasingly important role. Moreover, as the finite element software becomes more complex, the question of efficient simultaneous interfacing to multiple matrix solver packages matters. Every matrix solver comes with its own unique interface. Hardcoding this interface into a FEM solver means an unwanted coupling. If the FEM solver deals with multiple PDEs that produce matrices with substantially different properties (this is the case, e.g., with secondorder elliptic PDEs and Maxwell's equations), the application of multiple matrix solvers becomes a need.
Importance of parameter optimization in a nonlinear stabilized method adding a crosswind diffusion
Journal of Computational and Applied Mathematics, 2021
Abstract Numerical solutions of convection-dominated problems are known to exhibit spurious oscil... more Abstract Numerical solutions of convection-dominated problems are known to exhibit spurious oscillations whose suppression requires the use of numerical stabilization. Stabilized methods which involve heuristic parameters are often applied. The parameters influence the quality of the solution but their optimal values are unknown. In this paper, we consider a stabilization method which adds numerical diffusion adaptively, based on minimizing a functional. The novelty of our approach consists in combining an error indicator with reduced residuals with a nonlinear SOLD method adding artificial diffusion in the crosswind direction. We demonstrate that this approach can lead to more physically meaningful solutions than techniques considered before.
Advances in Applied Mathematics and Mechanics, 2010
Great algorithmic difficulty of hp-adaptive algorithms is one of the main obstacles preventing ad... more Great algorithmic difficulty of hp-adaptive algorithms is one of the main obstacles preventing adaptive hp-FEM from being employed widely in realistic engineering computations. In order to reduce their complexity, we present a new technique of arbitrary-level hanging nodes that eliminates forced refinements. By forced refinements we mean refinements which are not based on a large value of an error indicatior, but which are needed to keep the mesh sufficiently regular. The algorithmic treatment of forced refinements is highly problematic due to their recursive nature, and obviously, they slow down the performance of automatic adaptivity. We show that in the absence of forced refinements, the complexity of hp-adaptive algorithms drops dramatically while their efficiency improves. This is illustrated on a pair of numerical examples: an inner layer problem with known exact solution, and a problem related to microwave heating.
On A Mesh Generation Technique Based On A Special Smoothing Procedure For Uniform Inner Point Distribution
This paper is devoted to the triangular mesh generation on domains with arbitrary geometries. A s... more This paper is devoted to the triangular mesh generation on domains with arbitrary geometries. A special approach for the generation of an optimal number of uniformly distributed inner grid points is proposed. The scheme is based on an analogy with the construction of an equilibrium state of a system of a nite number of uniformly charged electrical particles closed in a bounded vessel. The method has minimum geometrical constraints and is therefore suitable also for domains with really complicated geometries. Ecient meshing procedure of the advancing front type is applied to the pre-constructed set of inner grid points and its reliability proved. An object-oriented freeware C++ library XGEN based on the presented method is developed. Advantages of the object-oriented approach are explained and the usage of the meshing tool briey described. Examples on the mesh generation on various domains are presented. 1 Introduction There is a number of various techniques for the 2D an...
Moving Particle Scheme for Triangular Grid Generation
Introduction There are many papers dealing with the 2D and 3D mesh generation based on a number o... more Introduction There are many papers dealing with the 2D and 3D mesh generation based on a number of various techniques (see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]). In this contribution we introduce a technique based on the application of an Advancing front algorithm to a set of optimally distributed internal grid points, obtained by the computer simulation of the behaviour of a special electrical system. Let us consider a bounded polygonal domain\Omega ae IR 2 with positive oriented boundary. Let us further consider a set GB ae @\Omega of boundary grid points P i , 1 i MB , dividing @\Omega into a set A ae @\Omega of disjoint abscissas A j , 1 j MB so that all the abscissas A j have the length h<F7.9
Three-dimensional Euler equations and Their Numerical Solution, Moving Particle Scheme for Grid Generation
icine, dealing with ow of blood. Part I of this thesis is devoted to the three-dimensional Euler ... more icine, dealing with ow of blood. Part I of this thesis is devoted to the three-dimensional Euler equations. First, we introduce some basic notation and physical quantities describing motion of uids. We will formulate the system of conservation laws which govern compressible inviscid ow. After having introduced the Finite Volume Method, prepared some basic theoretical tools and discussed the boundary conditions, we will develop four methods for solving it. These methods will be compared and then some more interesting numerical experiments performed. There is a wide class of literature concerning the study of numerical solution of the Euler equations which we think it is useful to mention here shortly { e.g. [13], [26], [31] [37], [36], [35], [7], [14], [33], [3], [5], [6], [11], [24], [27], [28], [23], [32] etc. In the second part we will deal with grid generation. We will set up a method based on computer simulation of a real physical process { interaction of elementary particles wi
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