We present an algorithm for fully automatic hp-adaptivity for finite element approximations of elliptic and Maxwell boundary value problems in three dimensions. The algorithm automatically generates a sequence of coarse grids, and a corresponding sequence of fine grids, such that the energy norm of error decreases exponentially with respect to the number of degrees of freedom in either sequence. At each step, we employ a discrete optimization algorithm to determine the refinements for the current coarse grid such that the projection-based interpolation error for the current fine grid solution decreases with an optimal rate with respect to the number of degrees of freedom added by the refinement. The refinements are restricted only by the requirement that the resulting mesh is at most 1-irregular, but they may be anisotropic in both element size h and order of approximation p. While we cannot prove that our method converges at all, we present numerical evidence of exponential convergence for a diverse suite of model problems from acoustic and electromagnetic scattering. In particular we show that our method is well suited to the automatic resolution of exterior problems truncated by the introduction of a perfectly matched layer. To enable and accelerate the solution of these problems on commodity hardware, we include a detailed account of three critical aspects of our implementations, namely an efficient implementations of sum factorization, several interfaces to the direct multi-frontal solver MUMPS, and some fast direct solvers for the computation of a sequence of nested projections. / text

No description available.

No description available.

No description available.

Thesis (M.Sc. (Dept. of Mathematics, Faculty of Natural Sciences))--University of the Western Cape, 2009. / Bibliography: leaves 180-195.

Numerical relativity is concerned with the generation of solutions to Einstein's equations by numerical means. In general, the construction of such a spacetime is accomplished in two stages: 1) the determination of initial data which is specified on a single spacelike hypersurface and satisfies four initial value equations, and 2) the evolution of the initial data to generate the spacetime or some portion of it. One of the key problems is the development of efficient algorithms for the solutions of these equations, as they are sufficiently complex to tax the fastest present computers. This thesis presents a comparison of various algorithms for the solution of the initial value equations, concentrating on the recently developed multi-grid method. The specific problem examined has been previously studied by Bowen, Piran and York. Their initial data has been interpreted as representing "snapshots" of three new families of black holes. Three of the four initial value equations possess analytic solutions. The remaining 2-dimensional nonlinear partial differential equation is solved numerically in this thesis using finite difference techniques. The performance of the multi-grid method, with respect to three more well-known methods, is evaluated through numerical experiments. The speed and reliability of the multi-grid algorithm are found to be very good. In addition, the results which had been previously calculated numerically by Piran are essentially reproduced, with the correction of some errors in that work. Possible extensions of the work to more complex initial value problems are also discussed. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

The bending of thin flat plates has occupied the interests of mechanicians and applied mathematicians since J. L. Lagrange discovered the differential equation characterizing the behavior of such structural members. One particular phase of investigation in this field concerns itself with the solution of the differential equation subject to given boundary conditions. Indeed, it may be safely stated that the bulk of the literature on the subject of flat plates is concerned with the solution of problems involving the specification of the transverse loading on the plate and the conditions at the boundary of the plate. Various mathematical techniques are available for the solution of such problems. Among these, the most prominent are, a.) the method of series, b) the method of singularities, and c) the complex variable techniques. A survey of the literature in this area has revealed a paucity of solutions of certain types of problems; notably, those problems in which boundary conditions are mixed a.long a portion of the edge of the plate which ha.s a continuously turning tangent. By mixed boundary conditions, we mean a. change in condition from prescription of bending moment and vertical shear to assignment of slope and deflection along a portion of the edge which has a continuously turning tangent. In the first section of this thesis, a number of problems are considered for the half-plane. The attendant boundary conditions considered a.re combinations of clamping and simple support. The second portion consists of a number of problems associated with the quarter-plane. Solutions for these problems are obtained by utilizing the method of images in conjunction with the solutions presented in the first section. After this, we examine some problems connected with the circular plate. In particular, a numerical solution is given for a uniformly loaded circular plate simply-supported over half of its boundary and clamped over the remaining portion. The last chapter is a brief discussion of plates in the form of rectangles. Here, a closed solution is presented for the bending moments in terms of Weierstrassian elliptic functions. Another numerical example is included for a uniformly loaded plate clamped over a portion of one edge and simply- supported over the remainder of its boundary. / Doctor of Philosophy

M.S.

M.S.

by Tam Yue Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 49-52). / Abstracts in English and Chinese. / List of Tables --- p.vi / List of Figures --- p.vii / Chapter 1 --- Solving Systems of ODEs with BVMs --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Background --- p.4 / Chapter 1.2.1 --- Linear Multistep Formulae --- p.4 / Chapter 1.2.2 --- Preconditioned GMRES Method --- p.6 / Chapter 1.3 --- Strang-Type Preconditioners with BVMs --- p.7 / Chapter 1.3.1 --- Block-BVMs and Their Matrix Forms --- p.8 / Chapter 1.3.2 --- Construction of the Strang-type Preconditioner --- p.10 / Chapter 1.3.3 --- Convergence Rate and Operation Cost --- p.12 / Chapter 1.3.4 --- Numerical Result --- p.13 / Chapter 1.4 --- Strang-Type BCCB Preconditioner --- p.15 / Chapter 1.4.1 --- Construction of BCCB Preconditioners --- p.15 / Chapter 1.4.2 --- Convergence Rate and Operation Cost --- p.17 / Chapter 1.4.3 --- Numerical Result --- p.19 / Chapter 1.5 --- Preconditioned Waveform Relaxation --- p.20 / Chapter 1.5.1 --- Waveform Relaxation --- p.20 / Chapter 1.5.2 --- Invertibility of the Strang-type preconditioners --- p.23 / Chapter 1.5.3 --- Convergence rate and operation cost --- p.24 / Chapter 1.5.4 --- Numerical Result --- p.25 / Chapter 1.6 --- Multigrid Waveform Relaxation --- p.27 / Chapter 1.6.1 --- Multigrid Method --- p.27 / Chapter 1.6.2 --- Numerical Result --- p.28 / Chapter 1.6.3 --- Concluding Remark --- p.30 / Chapter 2 --- Restoration of Chopped and Nodded Images --- p.31 / Chapter 2.1 --- Introduction --- p.31 / Chapter 2.2 --- The Projected Landweber Method --- p.35 / Chapter 2.3 --- Other Numerical Methods --- p.37 / Chapter 2.3.1 --- Tikhonov Regularization --- p.38 / Chapter 2.3.2 --- MRNSD --- p.41 / Chapter 2.3.3 --- Piecewise Polynomial TSVD --- p.43 / Chapter 2.4 --- Numerical Result --- p.46 / Chapter 2.5 --- Concluding Remark --- p.47 / Bibliography --- p.49