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The paper concerns numerical algorithms for solving the Beltrami equation fz¯(z) = μ(z)fz(z) for a compactly supported μ. First, we study an efficient algorithm that has been proposed in [P. Daripa, J. Comput. Phys., 106 (1993), pp. 355-365] and [P. Daripa and D. Mashat, Numer. Algorithms, 18 (1998), pp. 133-157] and present its rigorous justification. We then propose a different scheme for solving the Beltrami equation which has a comparable speed and accuracy, but has the virtue of easier implementation by avoiding the use of the Hilbert transform. The present paper can also be viewed as a prologue to one important application of the Beltrami equation: it provides a detailed description of the algorithm that has been used in [D. Gaidashev, Nonlinearity, 20 (1998), pp. 713-741] and [D. Gaidashev and M. Yampolsky, Experiment. Math., 16 (2007), pp. 215-226] to address an important issue in complex dynamics-conjectural universality for Siegel disks. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonoverlapping domain decomposition method for solving the three-dimensional Maxwell equations, based on the edge element discretization. For the Schur complement system on the interface, we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping sub domains. It is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we develop two least-squares approaches for the solution of the Stokes equations perturbed by a Laplacian term. (Such perturbed Stokes equations arise from finite element approximations of the Reissner­Mindlin plate.) Both are two-stage algorithms that solve first for the curls of the rotation of the fibers and the solenoidal part of the shear strain, then for the rotation itself (if desired). One approach uses L2 norms and the other approach uses H-1 norms to define the least-squares functionals. It is shown that the H-1 norm approach, under general assumptions, and the L2 norm approach, under certain H2 regularity assumptions, admit optimal performance for standard finite element discretization and either standard multigrid solution methods or pre-conditioners. These methods do not degrade when the perturbed parameter (the plate thickness) approaches zero. We also develop a three-stage least-squares method for the Reissner­Mindlin plate, which first solves for the curls of the rotation and the shear strain, next for the rotation itself, and then for the transverse displacement. [ABSTRACT FROM AUTHOR]

In this paper, we present a Gauss­Newton-based BFGS method for solving symmetric nonlinear equations which contain, as a special case, an unconstrained optimization problem, a saddle point problem, and an equality constrained optimization problem. A suitable line search is proposed with which the presented BFGS method exhibits an approximate norm descent property. Under appropriate conditions, global convergence and superlinear convergence of the method are established. The numerical results show that the proposed method is successful. [ABSTRACT FROM AUTHOR]

The mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains. The resulting methods are nonconforming but still yield optimal approximations. In this paper, we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric, second-order, elliptic equations in two dimensions. Both spectral and finite element methods, for geometrically conforming as well as nonconforming domain decompositions, are studied. In each case, we obtain a polylogarithmic bound on the condition number of the preconditioned matrix. [ABSTRACT FROM AUTHOR]

The combination technique is an algorithm for the approximate solution of partial differential equations on sparse grids that has to be combined with a suitable standard discretization. The advantage of the combination technique compared to the standard discretization is that the same accuracy is achieved with many fewer grid points. In this paper, the combination technique is used with a bilinear finite element discretization. Depending on the smoothness of the solution and the coefficients, it is proved for general second- order elliptic differential equations on the unit square that the combined solution converges with order O(h) or O(h log h-1) in the energy norm and with order O(h2 log h-1) or O(h 3⁄2 ) in the L2-norm, respectively. This holds even if the bilinear form corresponding to the elliptic equation is not symmetric positive definite. The proof does not use an asymptotic error expansion, but Sobolev space techniques. [ABSTRACT FROM AUTHOR]

A cell-centered finite volume method is proposed to approximate numerically the solution to the steady convection-diffusion equation on unstructured meshes of $d$-simplexes, where $d\geq 2$ is the spatial dimension. The method is formally second-order accurate by means of a piecewise linear reconstruction within each cell and at mesh vertices. An algorithm is provided to calculate nonnegative and bounded weights. Face gradients, required to discretize the diffusive fluxes, are defined by a nonlinear strategy that allows us to demonstrate the existence of a maximum principle. Finally, a set of numerical results documents the performance of the method in treating problems with internal layers and solutions with strong gradients. [ABSTRACT FROM AUTHOR]

A BDDC (balancing domain decomposition by constraints) method is developed for elliptic equations, with discontinuous coefficients, discretized by mortar finite element methods for geometrically nonconforming partitions in both two and three space dimensions. The coarse component of the preconditioner is defined in terms of one mortar constraint for each edge/face, which is the intersection of the boundaries of a pair of subdomains. A condition number bound of the form C maxi{(1 + log(Hi/hi))2} is established under certain assumptions on the geometrically nonconforming subdomain partition in the three-dimensional case. Here Hi and hi are the subdomain diameters and the mesh sizes, respectively. In the geometrically conforming case and the geometrically nonconforming cases in two dimensions, no assumptions on the subdomain partition are required. This BDDC preconditioner is also shown to be closely related to the Neumann-Dirichlet version of the FETI-DP algorithm. The results are illustrated by numerical experiments which confirm the theoretical results. [ABSTRACT FROM AUTHOR]

We present a new algorithm for numerically computing quadrature formulas for arbitrary domains which exactly integrate a given polynomial space. An effective method for constructing quadrature formulas has been to numerically solve a nonlinear set of equations for the quadrature points and their associated weights. Symmetry conditions are often used to reduce the number of equations and unknowns. Our algorithm instead relies on the construction of cardinal functions and thus requires that the number of quadrature points N be equal to the dimension of a prescribed lower dimensional polynomial space. The cardinal functions allow us to treat the quadrature weights as dependent variables and remove them, as well as an equivalent number of equations, from the numerical optimization procedure. We give results for the triangle, where for all degrees d ≤ 25, we find quadrature formulas of this form which have positive weights and contain no points outside the triangle. Seven of these quadrature formulas improve on previously known results. [ABSTRACT FROM AUTHOR]