This textbook may serve as a basis for an introductory course in particular for boundary element methods including modern trends such as fast boundary element methods and efficient solution methods, as well as the coupling of finite and boundary element methods. Finally, two chapters discuss advanced applications such as crack propagation or flow in fractured poroelastic media. The hp finite element discretizations, in particular, by spectral elements of elliptic equations are given significant attention in current research and applications. We also characterize a class of distributions of the coefficients, called quasi-monotone, for which the weighted L2-projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this short note we present new weighted Poincaré type inequalities for a class of piecewise constant coefficients that lead to sharper bounds independent of any possible large contrasts in the coefficients. Furthermore, we discuss implementational issues. Our theoretical findings are confirmed in numerical tests.
Author by : Mats G. This coefficient is allowed to have large jumps not only across but also along subdomain interfaces and in the interior of the subdomains. Author by : Carlos A. An attempt is made to develop robust methods for homogeneous and heterogeneous problems. This approach permits polyhedral element shapes as well as meshes with hanging nodes. We give a general theoretical result and then some numerical examples on a heterogeneous elasticity problem. We consider the problem of solving the algebraic system of equations which arise from the discretization of symmetric elliptic boundary value problems via finite element methods.
To be more precise, if the coefficient varies only moderately in a layer near the boundary of each subdomain, the method is proved to be robust with respect to arbitrary variation in the interior of each subdomain and with respect to coefficient jumps across subdomain interfaces. There are many domains with quite irregular boundaries, where these parameters stay bounded. The condition numbers are bounded uniformly in the mesh size h and the polynomial order p. Starting from Maxwell's equations, the mathematical model is derived, which is the variational formulation of a boundary value problem. The preconditioner is based on a three-level decomposition of the underlying ansatz space, the levels being piecewise bilinear functions on a coarse grid, piecewise bilinear functions on a fine grid, and piecewise polynomials of high degree on the fine grid.
The aim of this book is to provide an up to date and sound theoretical foundation for finite element methods in computational electromagnetism. It is aimed at researchers in applied mathematics, scientific computing or computational engineering. We derive sharp and explicit upper bounds for possibly weighted Poincaré constants of finite element stars. Although the aim of this book is to give a unified introduction into finite and boundary element methods, the main focus is on the numerical analysis of boundary integral and boundary element methods. The analysis is confirmed by numerical experiments for a model problem.
Außerdem ermöglichen sie die Kopplung verschiedener numerischer Verfahren und eine einfache Parallelisierung der Simulation zur Verkürzung der Rechenzeiten. The latter are star-shaped domains that consist of a finite number of nonoverlapping simplices or parallelepipeds which all share a common vertex. A new class of preconditioners for these discrete systems is developed based on substructuring also known as domain decomposition. The highly interdisciplinary content brings together contributions from scientists with complementary areas of expertise, such as pure and applied mathematicians, engineers, and biophysicists. The corresponding B-H-curve must in general be approximated from measurements. Aus Effizienzgründen wird das Einfachschichtpotential der Laplace-Gleichung als Vorkoditionierer verwendet und dessen Einsatz theoretisch abgesichert. Recently, Hofreither, Langer and Pechstein have analyzed a nonstandard finite element method based on element-local boundary integral operators.
Finally, numerical studies verify the efficiency of the Multigrid Newton method, tested on a model problem that simulates a magnetic valve. The paper is complemented with numerical results. Suitable variational formulations are developed and justified mathematically. The solution method for the considered nonlinear eigenvalue problem is based on the contour integral method, where special focus is put on the efficient numerical computation of the linear system along the boundary of the given search area. The purpose of this book is to give a detailed and self-contained presentation of these methods, including the corresponding algorithms as well as a rigorous convergence theory.
An efficient Full-Multigrid-Newton-Method is proposed. The text can serve also as a useful reference book and is intended for researchers, postgraduate students, and all practitioners working in the areas of efficient solution methods for partial differential equations. Additionally, the tearing and interconnecting framework is used to construct fast solvers for the resulting global system of equations. The book is for researchers and graduate students in applied mechanics and civil engineering. The proposed methods are applicable to problems on general domains involving differential operators with rather general coefficients. Please click button to get finite and boundary element tearing and interconnecting solvers for multiscale problems book now.
In order to get a good initial guess for the Newton iteration, we use the nested iteration strategy based on a hierarchy of nested grids. In this context, the book offers a detailed view to an active and up-to-date area of research. Examples throughout the book are intentionally kept simple so that the parallelization strategies are not dominated by technical details. Numerical results confirm the robustness of the proposed approach. When the diffusion coefficient varies within a subdomain or within a coarse grid element, then condition number bounds for these methods based on standard Poincaré inequalities may be overly pessimistic. However, due to the geometric partitioning of the update steps in this method, discontinuities in the solution appear that are physically implausible. In other cases, they are related to physical phenomena that actually take place on a very small scale but still have an important effect on the solution.
Finally, two chapters discuss advanced applications such as crack propagation or flow in fractured poroelastic media. The coarse level consists of the lowest order finite element space. In this context, the book offers a detailed view to an active and up-to-date area of research. Novel upscaling and preconditioning techniques, as well as applications to turbulent multi-phase flow, and to problems of current interest in materials science are all addressed. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial differential equations, especially for systems.