Abstract: This well written book is enlarged by the following topics: $B$-splines and their computation, elimination methods for large sparse systems of linear equations, Lanczos algorithm for eigenvalue problems, implicit shift techniques for the $LR$ and $QR$ algorithm, implicit differential equations, differential algebraic systems, new methods for stiff differential equations, preconditioning techniques and convergence rate of the conjugate gradient algorithm and multigrid methods for boundary value problems. Cf. also the reviews of the German original editions.

Abstract: Poisson surface reconstruction creates watertight surfaces from oriented point sets. In this work we extend the technique to explicitly incorporate the points as interpolation constraints. The extension can be interpreted as a generalization of the underlying mathematical framework to a screened Poisson equation. In contrast to other image and geometry processing techniques, the screening term is defined over a sparse set of points rather than over the full domain. We show that these sparse constraints can nonetheless be integrated efficiently. Because the modified linear system retains the same finite-element discretization, the sparsity structure is unchanged, and the system can still be solved using a multigrid approach. Moreover we present several algorithmic improvements that together reduce the time complexity of the solver to linear in the number of points, thereby enabling faster, higher-quality surface reconstructions.

Abstract: 1. A Simple Model Problem 2. Implementation 3. Auxiliary Results 4. Linear Elliptic Equations 5. Nonlinear Elliptic Equations 6. Parabolic Equations

Abstract: This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations.In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox.We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.

Abstract: We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms.

Abstract: What are Partial Differential Equations?.- Linear and Nonlinear Waves.- Fourier Series.- Separation of Variables.- Finite Differences.- Generalized Functions and Green's Functions.- Complex Analysis and Conformal Mapping.- Fourier Transforms.- Linear and Nonlinear Evolution Equations.- A General Framework for Linear Partial Differential Equations.- Finite Elements and Weak Solutions.- Dynamics of Planar Media.- Partial Differential Equations in Space.

Abstract: In this paper, we make use of the relatively new analytical technique, the homotopy decomposition method (HDM), to solve a system of fractional nonlinear differential equations that arise in the model for HIV infection of CD4+ T cells and attractor one-dimensional Keller-Segel equations. The technique is described and illustrated with a numerical example. The reliability of HDM and the reduction in computations give HDM a wider applicability. In addition, the calculations involved in HDM are very simple and straightforward.

Abstract: Ordinary Differential Equations, Boundary Value Problems, Fourier Series, and the Introduction to Integral Equations First-Order Differential Equations Second-Order Differential Equations Systems of Differential Equations Boundary Value Problems for Second-Order ODE and Sturm-Liouville Theory Qualitative Methods and Stability of ODE Solutions Method of Laplace Transforms for ODE Integral Equations Series Solutions of ODEs and Bessel and Legendre Equations Fourier Series Partial Differential Equations Introduction to PDE One-Dimensional Hyperbolic Equations Two-Dimensional Hyperbolic Equations One-Dimensional Parabolic Equations Two-Dimensional Parabolic Equations Elliptic Equations Appendix 1: Eigenvalues and Eigenfunctions of One-Dimensional Sturm-Liouville Boundary Value Problem for Different Types of Boundary Conditions Appendix 2: Auxiliary Functions, w(x,t), for Different Types of Boundary Conditions Appendix 3: Eigenfunctions of Sturm-Liouville Boundary Value Problem for the Laplace Equation in a Rectangular Domain for Different Types of Boundary Conditions Appendix 4: A Primer on the Matrix Eigenvalue Problems and the Solution of the Selected Examples in Sec. 5.2 Appendix 5: How to Use the Software Associated with the Book Bibliography

Abstract: SUMMARY This paper describes a nonlinear, three-dimensional spectral collocation method for the simulation of the incompressible Navier–Stokes equations under the Boussinesq approximation, motivated by geophysical and environmental flows. These flows are driven by the interaction of stratified fluid with topography, which this model accurately accounts for by using a mapped coordinate system. The spectral collocation method is implemented with both a Fourier trigonometric expansion and the Chebyshev polynomials, as appropriate for the domain boundary conditions. The coordinate mapping prohibits the use of existing, fast solution methods that rely on the separation of variables, so a preconditioner based on the approximate solution of a corresponding finite-difference problem with geometric multigrid is used. The model is parallelized with the Message Passing Interface library, and it runs effectively on shared and distributed-memory systems. Copyright © 2013 John Wiley & Sons, Ltd.

Abstract: We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments.

Abstract: A Local/global non-intrusive coupling algorithm is proposed for the analysis of mixed-mode crack propagation It is based on a three scale multigrid and extended finite element method, that was proposed recently for the direct estimation of stress intensity factors of static cracks The algorithm couples a linear elastic global model (possibly performed by a industrial software) with an enhanced local model capable of modeling a crack and accurately estimating SIFs (performed by a separate research code) It is said non-intrusive since it does not modify the global mesh, its connectivity and solver For the global model, the contribution of the local patch consists in additional nodal efforts near the crack, which makes it compatible with most softwares Further the shape of the domain over which the local model is applied is automatically adapted during propagation

Abstract: We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation mod- eling fluid flow. The CHB system is a generalizationof the Cahn-Hilliard-Stokesmodel and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step size. Owing to energy stability, we show that the scheme is stable in the time and space discrete l ¥ (0,T;H 1 h ) and l 2 (0,T;H 2 ) norms. We also present an efficient, practical non- linear multigrid method - comprised of a standard FAS method for the Cahn-Hilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part - for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy- and boundary-driven flows. AMS subject classifications: 65M06, 65M12, 65M55, 76T99

Abstract: A parallel geometric multigrid solver on hierarchically distributed grids is presented. Using a tree-structure for grid distribution onto the processing entities, the multigrid cycle is performed similarly to the serial algorithm, using additional vertical communication during transfer operations. The workload is gathered to fewer processes on coarser levels. Involved parallel structures are described in detail and the multigrid algorithm is formulated, discussing parallelization details. A performance study is presented that shows close to optimal efficiency for weak scaling up to 262k processes in 2 and 3 space dimensions.

Abstract: In this manuscript, we describe effective solvers for the optimal control of stabilized convectiondiffusion control problems. We employ the Local Projection Stabilization, which results in the same matrix system whether the discretize-then-optimize or optimize-then-discretize approach for this problem is used. We then derive two effective preconditioners for this problem, the first to be used with MINRES and the second to be used with the Bramble-Pasciak Conjugate Gradient method. The key components of both preconditioners are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to enact this latter approximation. We present numerical results to illustrate that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the step-size h and the regularization parameter ? for a range of problems.

Abstract: Poincare-type inequalities are a key tool in the analysis of partial differential equations. They play a particularly central role in the analysis of domain decomposition and multilevel iterative methods for second-order elliptic problems. When the diffusion coefficient varies within a subdomain or within a coarse grid element, then condition number bounds for these methods based on standard Poincare inequalities may be overly pessimistic. In this paper, we present new results on weighted Poincare-type inequalities for very general classes of coefficients that lead to sharper bounds independent of any possible large variation in the coefficients. The main requirement on the coefficients is some form of quasi-monotonicity that we will carefully describe and analyse. The Poincare constants depend on the topology and the geometry of regions of relatively high and/or low coefficient values, and we shall study these dependencies in detail. Applications of the inequalities in the analysis of domain decomposition and multigrid methods can be found in Pechstein & Scheichl (2011, Numer. Math., 118) and Scheichl et al. (2012, SIAM J. Numer. Anal., 50).

Abstract: In this paper we present a conservative numerical method for the Cahn-Hilliard equation with Dirichlet boundary conditions in complex domains. The method uses an unconditionally gradient stable nonlinear splitting numerical scheme to remove the high-order time-step stability constraints. The continuous problem has the conservation of mass and we prove the conservative property of the proposed discrete scheme in complex domains. We describe the implementation of the proposed numerical scheme in detail. The resulting system of discrete equations is solved by a nonlinear multigrid method. We demonstrate the accuracy and robustness of the proposed Dirichlet boundary formulation using various numerical experiments. We numerically show the total energy decrease and the unconditionally gradient stability. In particular, the numerical results indicate the potential usefulness of the proposed method for accurately calculating biological membrane dynamics in confined domains.

Abstract: ix

Abstract: This book introduces and analyzes the multigrid approach for the numerical solution of large sparse linear systems arising from the discretization of elliptic partial differential equations. Special attention is given to the powerful matrix-based-multigrid approach, which is particularly useful for problems with variable coefficients and nonsymmetric and indefinite problems. This approach applies not only to model problems on rectangular grids but also to more realistic applications with complicated grids and domains and discontinuous coefficients. Matrix-Based Multigrid can be used as a textbook in courses in numerical analysis, numerical linear algebra, and numerical PDEs at the advanced undergraduate and graduate levels in computer science, math, and applied math departments. The theory is written in simple algebraic terms and therefore requires preliminary knowledge in basic linear algebra and calculus only. Because it is self contained and includes useful exercises, the book is also suitable for self study by research students, researchers, engineers, and others interested in the numerical solution of partial differential equations.

Abstract: SUMMARY In this paper, we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of shifted Laplacian preconditioners is known to significantly speed up Krylov convergence. However, these preconditioners have a parameter β∈R, a measure of the complex shift. Because of contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter, which is predicted by a rigorous k-grid local Fourier analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge and being near optimal in terms of Krylov iteration count. Copyright © 2013 John Wiley & Sons, Ltd.

Abstract: Obstacle problems can be posed as elliptic variational inequalities, complementarity inequalities and Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs). In this paper, we propose a multigrid algorithm based on the full approximate scheme (FAS) for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A special coarse grid operator is proposed based on the Galerkin operator for the membrane constrained obstacle problem in this paper. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. Due to the nonlinear property of the minimal surface operator, the Galerkin operator for the minimal surface obstacle problem is not accurate. We introduce the direct discretization operator for the minimal surface obstacle problem. A special prolongation operator based on the bilinear interpolation is proposed to interpolate functions from the coarse grid to the fine grid. At the boundary between the active set and inactive set, the proposed prolongation operator can capture the active grid points and put accurate values at these points. We will demonstrate the fast convergence of the proposed multigrid method for solving obstacle problems by comparing with other multigrid methods.

Abstract: We consider anisotropic Allen--Cahn equations with interfacial energy induced by an anisotropic surface energy density $\gamma$. Assuming that $\gamma$ is positive, positively homogeneous of degree one, strictly convex in tangential directions to the unit sphere, and sufficiently smooth, we show stability of various time discretizations. In particular, we consider a fully implicit and a linearized time discretization of the interfacial energy combined with implicit and semi-implicit time discretizations of the double-well potential. In the semi-implicit variant, concave terms are taken explicitly. The arising discrete spatial problems are solved by globally convergent truncated nonsmooth Newton multigrid methods. Numerical experiments show the accuracy of the different discretizations. We also illustrate that pinch-off under anisotropic mean curvature flow is no longer frame invariant, but depends on the orientation of the initial configuration.