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In recent years, a nonoverlapping Robin-type domain decomposition method (DDM) for the finite element discretization systems of the second order elliptic equations, which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces, has been developed and analyzed since it was first proposed by P. L. Lions in [On the Schwarz alternating method III: A variant for nonoverlapping subdomains, in Proceedings of the 3rd International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1990, pp. 202-223]. However, the convergence rate of this DDM with many subdomains remains open when the lower term of equations vanishes. This open problem will be considered in this paper. The convergence rate is almost 1 — O(h¹/²H-1/²) in certain cases—for example, the case of a small number of subdomains, where h is the mesh size and H is the size of subdomain. In order to get the desirous convergence results, two mathematics skills are introduced in this paper; one is complexification of real linear space and the other is the spectral radius formula. [ABSTRACT FROM AUTHOR]

This paper presents a Lie-Trotter splitting for inertial Langevin equations (geometric Langevin algorithm) and analyzes its long-time statistical properties. The splitting is defined as a composition of a variational integrator with an Ornstein-Uhlenbeck flow. Assuming that the exact solution and the splitting are geometrically ergodic, the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial Langevin equations to within the accuracy of the variational integrator in representing the Hamiltonian. In particular, if the variational integrator admits no energy error, then the method samples the invariant measure of inertial Langevin equations without error. Numerical validation is provided using explicit variational integrators with first-, second-, and fourth-order accuracy. [ABSTRACT FROM AUTHOR]

This paper studies L² norm error estimates for the div least-squares method for which the associated homogeneous least-squares functional is equivalent to the H(div) × H¹ norm for the respective dual and primal variables. Least-squares of this type for the second-order elliptic equations, elasticity, and the Stokes equations are an active area of research, and error estimates in the H(div) × H¹ norm were previously established. In this paper, we establish optimal L² norm error estimates for the primal variable under the minimum regularity requirement through a refined duality argument. [ABSTRACT FROM AUTHOR]

Pseudo-transient continuation is a Newton-like iterative method for computing steady-state solutions of differential equations in cases where the initial data are far from a steady state. The iteration mimics a temporal integration scheme, with the time step being increased as steady state is approached. The iteration is an inexact Newton iteration in the terminal phase. In this paper we show how steady-state solutions to certain ordinary and differential algebraic equations with nonsmooth dynamics can be computed with the method of pseudo-transient continuation. An example of such a case is a discretized PDE with a Lipschitz continuous, but nondifferentiable, constitutive relation as part of the nonlinearity. In this case we can approximate a generalized derivative with a difference quotient. The existing theory for pseudo-transient continuation requires Lipschitz continuity of the Jacobian. Newton-like methods for nonsmooth equations have been globalized by trust-region methods, smooth approximations, and splitting methods in the past, but these approaches are not designed to find steady-state solutions of time-dependent problems. The method in this paper synthesizes the ideas from nonsmooth calculus and the method of pseudo-transient continuation. [ABSTRACT FROM AUTHOR]

In this paper, we develop a C0 interior penalty method for a fourth order singular perturbation elliptic problem in two dimensions on polygonal domains. Using some a posteriori error analysis techniques, we are able to show that the method converges in the energy norm uniformly with respect to the perturbation parameter under minimal regularity assumptions. In addition, we analyze the convergence of the numerical solution to the unperturbed second order problem. Finally, we perform some numerical experiments that back up the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a uniaxial perfectly matched layer (PML) method for solving the tilne-harnlonic scattering problems ill two-layered media. The exterior region of the scatterer is divided into two half spaces by all infinite plane, on two sides of which the wave number takes different vahms. We surround the conlputational donmin where the scattering field is interested by a PML with the uniaxial medium property. By imposing homogeneous boundary condition on the outer boundary of the PML, we show that the solution of the PML problem converges exponentially to the solution of the original scattering problem in the computational dolnain as either the PML absorbing coefficient or the thickness of the PML tends to infinity. [ABSTRACT FROM AUTHOR]

This paper continues the convergence analysis in [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882-904] of discrete approximations to the linearized and adjoint equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. We consider a simple modified Lax-Friedrichs discretization on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It is proved that there is convergence in the discrete approximation of linearized output functionals even for Dirac initial perturbations and pointwise convergence almost everywhere for the solution of the adjoint discrete equations. In particular, the adjoint approximation converges to the correct uniform value in the region in which characteristics propagate into the discontinuity. Moreover, it is shown that the results of [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882-904] and the present paper hold also for quite general nonlinear initial data which contain multiple shocks and for which shocks form at a later time and/or merge. [ABSTRACT FROM AUTHOR]

This paper studies a new recovery-based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems. Instead of recovering the gradient in the continuous finite element space, the flux is recovered through a weighted L2 projection onto H(div) conforming finite element spaces. The resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results. This paper also proposes an adaptive finite element method based on either the recovery-based estimators or the edge estimator through local mesh refinement and establishes its convergence. In particular, it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps. [ABSTRACT FROM AUTHOR]

A full discrete two-level postprocessing Fourier Galerkin scheme for the unsteady Navier-Stokes equations with a periodic boundary condition is proposed in this paper. By defining a new projection, the interaction between the large and small eddies is reflected by the associated space splitting to some extent. Therefore, a weakly coupled system of the large and small eddies is obtained. Stability and error estimates for the weakly coupled two-level scheme are established in the paper. The proposed scheme is an effective algorithm because nonlinearity is treated only in the coarse-level subspace by solving the coarse-level standard Galerkin equation, and the matrix of the linear algebraic equations arising in each fine-level time stepping is a sparse matrix, especially for large scale computations. As a result, the new scheme saves a lot of CPU time and memory in comparison with the standard Fourier Galerkin method for deriving an approximation of prescribed accuracy. [ABSTRACT FROM AUTHOR]

In this paper we consider numerical approximations of a constraint minimization problem, where the object function is a quadratic Dirichlet functional for vector fields and the interior constraint is given by a convex function. The solutions of this problem are usually referred to as harmonic maps. The solution is characterized by a nonlinear saddle point problem, and the corresponding linearized problem is well-posed near strict local minima. The main contribution of the present paper is to establish a corresponding result for a proper finite element discretization in the case of two space dimensions. Iterative schemes of Newton type for the discrete nonlinear saddle point problems are investigated, and mesh independent preconditioners for the iterative methods are proposed. [ABSTRACT FROM AUTHOR]

This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Ampère equation det(D²u0) = f (> 0) based on the vanishing moment method which was proposed recently by the authors in [X. Feng and M. Neilan, J. Scient. Comp., DOI 10.1007/s10915-008-9221-9, 2008]. In this approach, the second-order fully nonlinear Monge-Amp`ere equation is approximated by the fourth order quasilinear equation -ϵΔ²uϵ + detD²uϵ = f. It was proved in [X. Feng, Trans. AMS, submitted] that the solution ue converges to the unique convex viscosity solution u0 of the Dirichlet problem for the Monge-Ampère equation. This result then opens a door for constructing convergent finite element methods for the fully nonlinear second-order equations, a task which has been impracticable before. The goal of this paper is threefold. First, we develop a family of Hermann-Miyoshi-type mixed finite element methods for approximating the solution uϵ of the regularized fourth-order problem, which computes simultaneously ue and the moment tensor σϵ := D²uϵ . Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter ϵ, for the errors uϵ - uϵ h and σ0 - σϵ h . Finally, we present a detailed numerical study on the rates of convergence in terms of powers of e for the error u0 -uϵ h and σϵ -σϵh , and numerically examine what is the "best" mesh size h in relation to e in order to achieve these rates. Due to the strong nonlinearity of the underlying equation, the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem. To overcome the difficulty, we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations. [ABSTRACT FROM AUTHOR]

In their 2006 paper, Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. prove that a sufficiently smooth strong solution of the 3D Navier-Stokes equations is robust with respect to small enough changes in initial conditions and forcing function. They also show that if a regular enough strong solution exists, then Galerkin approximations converge to it. They then use these results to conclude that the existence of a sufficiently regular strong solution can be verified using sufficiently refined numerical computations. In this paper we study the strong solutions with less regularity than those considered in Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. We prove a similar robustness result and show the validity of the results relating convergent numerical computations and the existence of the strong solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence properties of iterative algorithms of conjugate gradient type applied to the computation of an extreme eigenvalue of a Hermitian operator via the optimization of the Rayleigh quotient functional. The algorithms in focus employ the line search for the extremum of the Rayleigh quotient functional in the direction that is a linear combination of the gradient and the previous search direction. An asymptotically explicit equation for the reduction in the eigenvalue error after the line search step as a function of the search direction is derived and applied to the analysis of the local convergence properties (i.e., the error reduction after an iteration) of the algorithms in question. The local (stepwise) asymptotic equivalence of various conjugate gradient algorithms and the generalized Davidson method is proved, and a new convergence estimate for conjugate gradient iterations is derived, showing the reduction in the eigenvalue error after any two consecutive iterations. The paper's analysis extensively employs remarkable properties of the operator of the so-called Jacobi orthogonal complement correction equation. [ABSTRACT FROM AUTHOR]

It is known that some traditional shock-capturing schemes, including Godunov- or Roe-type schemes, generate downstream oscillatory wave patterns that cannot be effectively dampened by the dissipation methods they provide. The theme of this paper is to understand the formation and behavior at these upstream/downstream patterns from the point view of the symmetry inherent in the PDE. This paper investigates symmetry breaking and spatial oscillations in finite difference solutions for the inviscid Burgers and one-dimensional Navier-Stokes shock equations using the Lax-Wendroff, TVD, and nonoscillatory, nonfree parameter dissipative schemes. Symmetry breaking is considered as a candidate for the inherent cause of the nonphysical oscillations in the vicinity of the shock. [ABSTRACT FROM AUTHOR]

The paper is concerned with convergence estimates for the preconditioned steepest descent method for the computation of the smallest eigenvalue of a Hermitian operator. Available estimates are reviewed and new estimates are introduced that improve on the known ones in certain respects. In addition to the estimates for the error reduction after one iteration, we consider estimates for the so-called asymptotic convergence factor defined as the upper limit of the average error reduction per iteration. The paper focuses on sharp estimates, i.e., those that cannot be improved without using additional information. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze explicit exponential Runge--Kutta methods for the time integration of semilinear parabolic problems. The analysis is performed in an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities. We commence by giving a new and short derivation of the classical (nonstiff) order conditions for exponential Runge--Kutta methods, but the main interest of our paper lies in the stiff case. By expanding the errors of the numerical method in terms of the solution, we derive new order conditions that form the basis of our error bounds for parabolic problems. We show convergence for methods up to order four, and we analyze methods that were recently presented in the literature. These methods have classical order four, but they do not satisfy some of the new conditions. Therefore, an order reduction is expected. We present numerical experiments which show that this order reduction in fact arises in practical examples. Based on our new conditions, we finally construct methods that do not suffer from order reduction. [ABSTRACT FROM AUTHOR]

Over the last few years a great effort has been made to develop monotone high order explicit Runge--Kutta methods by means of their Shu--Osher representations. In this context, the stepsize restriction to obtain numerical monotonicity is normally computed using the optimal representation. In this paper we extend the Shu--Osher representations for any Runge--Kutta method giving sufficient conditions for monotonicity. We show how optimal Shu--Osher representations can be constructed from the Butcher tableau of a Runge--Kutta method. The optimum stepsize restriction for monotonicity is given by the radius of absolute monotonicity of the Runge--Kutta method [L. Ferracina and M. N. Spijker, SIAM J. Numer. Anal., 42 (2004), pp. 1073--1093], and hence if this radius is zero, the method is not monotone. In the Shu--Osher representation, methods with zero radius require negative coefficients, and to deal with them, an extra associate problem is considered. In this paper we interpret these schemes as representations of perturbed Runge--Kutta methods. We extend the concept of radius of absolute monotonicity and give sufficient conditions for monotonicity. Optimal representations can be constructed from the Butcher tableau of a perturbed Runge--Kutta method. [ABSTRACT FROM AUTHOR]

We consider the scalar conservation law with flux function discontinuous in the space variable, i.e., \begin{eqnarray} \label{eq1} u_t+(H(x)f(u)+(1-H(x))g(u))_{x} &=& 0 \quad \mbox{in } \R \times \R_{+}, \nonumber \\ u(0, x) &=& u_{0}(x) \quad \mbox{in } \R, \label{0.1} \end{eqnarray} where $H$ is the Heaviside function and $f$ and $g$ are smooth with the assumptions that either $f$ is convex and $g$ is concave or $f$ is concave and $g$ is convex. The existence of a weak solution of (\ref{eq1}) is proved by showing that upwind finite difference schemes of Godunov and Enquist--Osher type converge to a weak solution. Uniqueness follows from a Kruzkhov-type entropy condition. We also provide explicit solutions to the Riemann problem for (\ref{eq1}). At the level of numerics, we give easy-to-implement numerical schemes of Godunov and Enquist--Osher type. The central feature of this paper is the modification of the singular mapping technique (the main analytical tool for these types of equations) which allows us to show that the numerical schemes converge. Equations of type (\ref{eq1}) with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux: \begin{equation} \label{eq2} \begin{array}{r@{\;}l} u_t + (k (x) f (u))_x &= 0, \\ u (0, x) &= u_0 (x), \end{array} \end{equation} with $f$ convex and $k$ of indefinite sign. [ABSTRACT FROM AUTHOR]

This paper provides a theoretical analysis of a higher-order, FFT-based integral equation method introduced recently [IEEE Trans. Antennas and Propagation, 48 (2000), pp. 1862-1864] for the evaluation of transverse electric-polarized electromagnetic scattering from a bounded, penetrable inhomogeneity in two-dimensional space. Roughly speaking, this method is based on Fourier smoothing of the integral operator and the refractive index n (x). Here we prove that the solution of the resulting integral equation approximates the solution of the exact integral equation with higher-order accuracy, even when n (x) is a discontinuous function-as suggested by the numerical experiments contained in the paper mentioned above. In detail, we relate the convergence rates of the computed interior and exterior fields to the regularity of the scatterer, and we demonstrate, with a few numerical examples, that the predicted convergence rates are achieved in practice. [ABSTRACT FROM AUTHOR]

This is the third part of a series of papers on least-squares Galerkin methods for parabolic initial-boundary value problems. These methods are based on the minimization of a least-squares functional for an equivalent first-order system over space and time with respect to suitable discrete spaces. This paper presents the derivation and analysis of one-step methods for semidiscretization in time from least-squares principles for semilinear parabolic problems. One of the most important features of the least-squares methodology is a built-in a posteriori estimate for the approximation error. For the presentation in this paper, we focus our attention on the specific combination of piecewise linear, not necessarily continuous, functions in time with continuous piecewise linear for the flux and scalar variables, respectively. For the resulting method, a convergence result is shown for the scalar variable. [ABSTRACT FROM AUTHOR]

The paper presents a new affine invariant theory on asymptotic mesh independence of Newton's method for discretized nonlinear operator equations. Compared to earlier attempts, the new approach is both much simpler and more intuitive from the algorithmic point of view. The theory is exemplified at finite element methods for elliptic PDE problems. [ABSTRACT FROM AUTHOR]

In this paper, a local minimax-Newton method is developed to solve for multiple saddle points. The local minimax method [SIAM I Sci. Comput., 23 (2001), pp. 840-865]. is used to locate an initial guess and a version of the generalized Newton method is used to speed up convergence. When a problem possesses a symmetry, the local minimax method is invariant to the symmetry. Thus the symmetry can be used to greatly enhance the efficiency and stability of the local minimax method. But such an invariance is sensitive to numerical error and the Haar projection has been used to enforce the symmetry [SIAM I Numer. Anal., submitted]. In this paper, we prove that the Newton method is invariant to symmetries and that such an invariance is insensitive to numerical error. When a symmetric degeneracy takes place, it is proved that the Newton direction can be easily solved in an invariant subspace. Thus the Newton method can be used not only to speed up convergence but also to avoid using the Haar projection if the symmetric degeneracy is removable by a discretization. Finally, numerical examples are presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

This paper discusses convergence behavior of the nonlinear inexact Uzawa algorithm for solving saddle-point problems presented in a recent paper of Bramble, Pasciak, and Vassilev (SIAM J. Numer. Anal., 34 (1997), pp. 1072­1092). It is shown that this algorithm converges under a condition weaker than that stated in their paper. Key words: saddle-point problems; nonlinear inexact Uzawa algorithm; convergence behavior [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of implicit neutral functional differential equations. Based on the continuous Runge­Kutta method (for ordinary differential equations) and the collocation method (for functional equations), two general one-step methods are formulated and their uniform order of approximation is discussed. Numerical stability of a class of Runge­Kutta-collocation methods is analyzed. In the case where a neutral equation can be expressed in an explicit as well as an implicit form, the advantage of constructing a numerical scheme directly from the implicit form is explained theoretically and demonstrated by a numerical example. [ABSTRACT FROM AUTHOR]

Liu, Osher, and Chan introduced weighted essentially nonoscillatory (WENO) reconstructions in [X.-D. Liu, S. Osher, and T. Chan, J. Comput. Phys., 115 (1994), pp. 200-212] to improve the order of accuracy of essentially nonoscillatory (ENO) reconstructions [A. Harten et al., J. Comput. Phys., 71 (1987), pp. 231-303]. In [G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), pp. 202-228], the authors proposed smoothness indicators to obtain a WENO fifth order reconstruction from third order ENO reconstructions. With these smoothness indicators, Balsara and Shu [J. Comput. Phys., 160 (2000), pp. 405-452] and, later, [G. A. Gerolymos, D. Sénéchal, and I. Vallet, J. Comput. Phys., 228 (2009), pp. 8481-8524] obtained (2r - 1)th order WENO recoilstructions from rth order ENO reconstructions for 4 < r < 6, resp., 7 < r < 9. In [A. K. Henrick, T. D. Aslam, and J. M. Powers, J. Comput. Phys., 207 (2005), pp. 542-567], the authors noticed that these reconstructions do not attain the optimal order 2r - 1 at extrema and they proposed a fix for the problem. Other authors [R. Borges et al., J. Comput. Phys., 227 (2008), pp. 3191-3211; N. K. Yamaleev and M. H. Carpenter, d. Comput. Phys., 228 (2009), pp. 4248-4272] have addressed this problem with different weight designs for Jiang-Shu smoothness indicators. In this paper we exploit the special structure of Jiang-Shu smoothness indicators and analyze the role of a parameter appearing in the weight definition to avoid division by zero to obtain for any r > 2, by standard approximation properties of Lagrange interpolation, that the order of the WENO reconstruction is 2r - 1 at smooth regions, regardless of neighboring extrema, whilst this order is r, as ENO reconstructions have, when the function has a discontinuity in the stencil of 2r - 1 points but it is smooth in at least one of the substencils of r points. The optimal weights are also obtained in closed form. [ABSTRACT FROM AUTHOR]

We study a block triangular preconditioner for finite element approximations of the linearized Navier-Stokes equations. The preconditioner is based on the augmented Lagrangian formulation of the problem and was introduced by the authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095-2113]. In this paper we prove field-of-values type estimates for the preconditioned system which lead to optimal convergence bounds for the GMRES algorithm applied to solve the system. Two variants of the preconditioner are considered: an ideal one based on exact solves for the velocity submatrix, and a more practical variant based on block triangular approximations of the velocity submatrix. [ABSTRACT FROM AUTHOR]

A theoretical analysis of a first-order div least-squares finite element method is presented. Our main interest is providing L2-norm error estimates for the primary function u and the flux σ = -효∇u. While there has been significant progress in the error estimates for the flux in L2-norm, these estimates have drawbacks, such as requiring smooth solutions or that the error estimate for ||σ - σh||0 is coupled with the error estimate for ||u - uh||1, where (uh, σh) is the least squares approximate solution for (u, σ = - 효 ∇u). In this paper, with the minimum regularity assumption u ∈ H1+α, α > 1/2, we separate the error estimate for ||σ - ||h||0 from ||u - uh||1 and establish that ||σ - σh||0 is bounded by the best approximation for a in the finite element space and ||u - uhh||0. Then, we proceed to obtain two new estimates for ||u - uh||0. The first estimate is useful to understand how the accuracy of ||u - uh||0 affects the accuracy of ||σ - σh||0 and the other sheds new light on the convergence behavior of ||u - uh||0. Our analysis does not require smooth solutions and the domain is allowed to be nonconvex. The resulting error estimates are valid for both smooth and low regularity solutions and show that optimal rates of convergence can be achieved by choosing proper finite element spaces such as the Raviart Thomas spaces for the flux σ. [ABSTRACT FROM AUTHOR]

The paper presents a generalization of Arnold-Falk-Winther elements for linear elasticity to meshes with elements of variable order. The generalization is straightforward but the stability analysis involves a nontrivial modification of involved interpolation operators. The analysis addresses only the h-convergence. In addition, numerical experiments for two dimensional cases are provided. [ABSTRACT FROM AUTHOR]

This paper is concerned with delay-independent asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods. We first give some sufficient conditions for the delay-independent asymptotic stability of these equations. Then we study and derive a sufficient and necessary condition for the delay-independent asymptotic stability of numerical solutions obtained by linear multistep methods combined with Lagrange interpolation. Finally, one numerical example is performed to confirm our theoretical result. [ABSTRACT FROM AUTHOR]

First-order div least-squares finite element methods for second-order elliptic partial differential equations are considered. While there has been significant progress in error estimates for the methods, to the best of our knowledge these estimates are based on the global norm of the error. In this paper, we provide highly localized pointwise error estimates for the primary variable u by establishing that the least-squares solutions are locally higher-order perturbations of the standard Galerkin solutions. As elementary consequences, we identify the superconvergent points. The set of superconvergent points for the primary function of the least-squares solution is the same as that of the standard Galerkin solution. Also, we present a class of average-type a posteriori error estimators for the method, and conditions are given under which they are asymptotically exact or equivalent estimators in the maximum-norm on each single element of the underlying mesh. [ABSTRACT FROM AUTHOR]

In this paper, the ill-posed problem of computation of the Fourier transform f̂ for a noisy bandlimited signal f is discussed. Here, f̂ is computed by a regularized. Fourier series which is obtained by a regularization of the Fourier transformed sampling formula of Shannon. If the regularization parameter tends to zero, the uniform and L²-convergence of this method is proved. Numerical examples show the performance of this approach. [ABSTRACT FROM AUTHOR]

The coupled problem with Stokes flow in one subdomain and a Darcy flow model in a second subdomain is studied in this paper. Both flow problems are treated as first-order systems, involving pseudostress and velocity in the Stokes case and using a flux-pressure formulation in the Darcy subdomain as process variables, respectively. The Beavers-Joseph-Saffman interface conditions are treated by an appropriate interface functional which is added to the least squares functional associated with the subdomain problems. A combination of H(div)-conforming Raviart Thomas and standard H¹-conforming elements is used for the Stokes as well as for the Darcy subsystem. The homogeneous least squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates. It also establishes the fact that the local evaluation of the least squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies. [ABSTRACT FROM AUTHOR]

Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator - Δ - ω², ω > 0. They include the so-called ultra weak variational formulation from. This paper is concerned with the a priori convergence analysis of PWDG in the ease of p-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121-136], and plane wave approximation theory. [ABSTRACT FROM AUTHOR]

We present a simple and easy-to-implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, classical finite difference discretizations can be shown to converge to the unique viscosity solutions of the considered problems. However, especially when using fully implicit time stepping schemes with their desirable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to an order of O(1/ρ), where ρ > 0 is the penalty parameter, and we show that an iterative scheme can be used to solve the penalized discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results. [ABSTRACT FROM AUTHOR]

A set χN of N points on the unit sphere is a spherical t-design if the average value of any polynomial of degree at most t over χN is equal to the average value of the polynomial over the sphere. This paper considers the characterization and computation of spherical t-designs on the unit sphere 핊² ⊂ ℝ³ when N ≥ (t + 1)², the dimension of the space ℙt of spherical polynomials of degree at most t. We show how to construct well conditioned spherical designs with N ≥ (t + 1)² points by maximizing the determinant of a matrix while satisfying a system of nonlinear constraints. Interval methods are then used to prove the existence of a true spherical t-design very close to the calculated points and to provide a guaranteed interval containing the determinant. The resulting spherical designs have good geometrical properties (separation and mesh norm). We discuss the usefulness of the points for both equal weight numerical integration and polynomial interpolation on the sphere and give an example. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes some fully discrete (multiphysics) finite element methods for a displacement-pressure model which describes swelling dynamics of polymer gels under mechanical constraints. By introducing an "elastic pressure" we first present a reformulation of the original model. We then propose a time-stepping scheme which decouples the PDE system at each time step into two subproblems, one of which is a Stokes-like problem for the displacement vector field (of the solid network of the gel) and the other is a diffusion problem for a "pseudopressure" field (of the solvent of the gel). To make such a multiphysics approach feasible, it is vital to find admissible constraints to resolve the uniqueness issue for the Stokes-like problem and to construct a good boundary condition for the diffusion equation so that it also becomes uniquely solvable. The key to the first difficulty is to discover certain conserved quantities for the PDE solution, and the solution to the second difficulty is to use the Stokes-like problem to generate a boundary condition for the diffusion problem. The time-stepping scheme allows one to use any convergent Stokes solver (and its code) together with any convergent diffusion equation solver (and its code) to solve the polymer gel model. In the paper, the Taylor-Hood mixed finite element method combined with the continuous linear finite element method are chosen as an example to present the ideas and to demonstrate the viability of the proposed multiphysics approach. It is proved that, under a mesh constraint, both the semidiscrete (in space) and fully discrete methods enjoy some discrete energy laws which mimic the differential energy law satisfied by the PDE solution. Optimal order error estimates in various norms are established for both semidiscrete and fully discrete methods. Numerical experiments are also presented to show the performance of the proposed approach and methods. [ABSTRACT FROM AUTHOR]

This paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. A simple modified Lax-Friedrichs discretization is used on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It is proved that this gives pointwise convergence almost everywhere for the solution to the linearized discrete equations with smooth initial data, and also convergence in the discrete approximation of linearized output functionals. In Part 2 [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 905-921] we extend the results to Dirac initial data for the linearized equation and will prove the pointwise convergence almost everywhere for the solution of the adjoint discrete equations. In addition, we present numerical results illustrating the asymptotic behavior which is analyzed. [ABSTRACT FROM AUTHOR]

This paper takes another look at the convergence analysis of the Arnoldi procedure for solving non-Hermitian eigenvalue problems. Two main viewpoints are put in contrast. The first exploits the eigenbasis, when there is one, and relies on classical min-max approximation theory results. The second approach relies on the Schur factorization. Its aim is to link the convergence analysis of the Arnoldi process for eigenvalue problems to that of the generalized minimal residual iterations (GMRES), for which much is known. [ABSTRACT FROM AUTHOR]

Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian formulation of the problem, which leads to a notion of well posedness for inverse problems, at the level of probability measures. The stability which results from this well posedness may be used as the basis for quantifying the approximation, in finite dimensional spaces, of inverse problems for functions. This paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem. This is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem. It is noteworthy that, when the prior is a Gaussian random field model, controlling differences in the Hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators, including the mean and covariance operator. The ideas are applied to some non-Gaussian inverse problems where the goal is determination of the initial condition for the Stokes or Navier-Stokes equation from Lagrangian and Eulerian observations, respectively. [ABSTRACT FROM AUTHOR]

Convergence rates for finite element discretizations of elliptic eigenvalue problems in the literature usually are of the following form: If the mesh width h is fine enough, then the eigenvalues, resp., eigenfunctions, converge at some well-defined rate. In this paper, we will determine the maximal mesh width h0-more precisely the minimal dimension of a finite element space-so that the asymptotic convergence estimates hold for h ≤ h0. This mesh width will depend on the size and spacing of the exact eigenvalues, the spatial dimension, and the local polynomial degree of the finite element space. For example, in the one-dimensional case, the condition λ3/4h0 ≲ 1 is sufficient for piecewise linear finite elements to compute an eigenvalue λ with optimal convergence rates as h0 ≥ h → 0. It will turn out that the condition for eigenfunctions is slightly more restrictive. Furthermore, we will analyze the dependence of the ratio of the errors of the Galerkin approximation and of the best approximation of an eigenfunction on λ and h. In this paper, the error estimates for the eigenvalue/-function are limited to the selfadjoint case. However, the regularity theory and approximation property cover also the nonselfadjoint case and, hence, pave the way towards the error analysis of nonselfadjoint eigenvalue/-function problems. [ABSTRACT FROM AUTHOR]

In this paper we prove an optimal error estimate for the H(curl)-conforming projection-based p-interpolation operator introduced in (L. Demkowicz and I. Babuška, SIAM J. Numer. Anal., 41(2003), pp. 1195-1208]. This result is proved on the reference element (either triangle or square) K for regular vector fields in Hr (curl, K) with arbitrary r > 0. The formulation of the result in the H(div)-conforming setting, which is relevant for the analysis of high-order boundary element approximations for Maxwell's equations, is provided as well. [ABSTRACT FROM AUTHOR]

The need to evaluate expressions of the form f(A) or f(A)b, where f is a nonlinear function, A is a large sparse n × n matrix, and b is an n-vector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process. Applications of the Faber transform to rational approximation methods and, in particular, to the rational Arnoldi process also are discussed. [ABSTRACT FROM AUTHOR]

For Runge-Kutta methods (RKMs), linear multistep methods (LMMs), and classes of general linear methods (GLMs), much attention has been paid, in the literature, to special nonlinear stability requirements indicated by the terms total-variation-diminishing, strong stability preserving, and monotonicity. Stepsize conditions, guaranteeing these properties, were derived by Shu & Ocher [J. Comput. Phys., 77 (1988), pp. 439-471] and in numerous subsequent papers. These special stability requirements imply essential boundedness properties for the numerical methods, among which the property of being total-variation-bounded. Unfortunately, for many well-known methods, the above special requirements are violated, so that one cannot conclude in this way that the methods are (total-variation-)bounded. In this paper, we focus on stepsize conditions for boundedness directly, rather than via the detour of the above special stability properties. We present a generic framework for deriving best possible stepsize conditions which guarantee boundedness of actual RKMs, LMMs, and GLMs, thereby generalizing results on the special stability properties mentioned above. [ABSTRACT FROM AUTHOR]

Generalizing recent results in [M. K. Camlibel, Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2001], [M. K. Camlibel, W. P. M. H. Heemels, and J. M. Schumacher, IEEE Trans. Circuits Systems I: Fund. Theory Appl., 49 (2002), pp. 349-357], and [J.-S. Pang and D. Stewart, Math. Program. Ser. A, 113 (2008), pp. 345-424], this paper provides an in-depth analysis of time-stepping methods for solving initial-value and boundary-value, non-Lipschitz linear complementarity systems (LCSs) under passivity and broader assumptions. The novelty of the methods and their analysis lies in the use of "least-norm solutions" in the discrete-time linear complementarity subproblems arising from the numerical scheme; these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets. Among the principal results, it is shown that, using such least-norm solutions of the discrete-time subproblems, an implicit Euler scheme is convergent for passive initial-value LCSs; generalizations under a strict copositivity assumption and for boundary-value LCSs are also established. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a seminorm for a subspace. The optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces. On each subspace an iterative proximity-map algorithm is implemented via oblique thresholding, which is the main new tool introduced in this work. We provide convergence conditions for the algorithm in order to compute minimizers of the target energy. Analogous results are derived for a parallel variant of the algorithm. Applications are presented in domain decomposition methods for degenerate elliptic PDEs arising in total variation minimization and in accelerated sparse recovery algorithms based on l1-minimization. We include numerical examples which show efficient solutions to classical problems in signal and image processing. [ABSTRACT FROM AUTHOR]

In this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated. The main idea is to use finite element spaces that are induced by triangulations of an "outer" domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface. We give an analysis that shows that the method has optimal order of convergence both in the H¹- and in the L²-norm. Results of numerical experiments are included that confirm this optimality. [ABSTRACT FROM AUTHOR]

Certain rheological behaviors of fluids in engineering sciences are modeled by power law ansatz with p ∈ (1, 2]. In the present paper a semi-implicit time discretization scheme for such fluids is proposed. The main result is the optimal O(k) error estimate, where k is the time step size. Our results hold in the range p ∈ (3/2, 2] (in the three-dimensional setting) for strong solutions of the continuous problem, whose existence is guaranteed under appropriate assumptions on the data. The estimates are uniform with respect to the degeneracy parameter δ ∈ [0, δ0] of the extra stress tensor. Additional regularity properties of the solution of the discrete problem are proved. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the error estimate of the bilinear interpolation on a general quadrilateral and the geometric characters of the quadrilateral. Some explicit bounds of the interpolation error are obtained based on some sharp estimates of the integral over 1/∣J∣p-1 for 1 ≤ p≤∞ on the reference element, where J is the Jacobian of the nonaffine mapping. This allows us to introduce weak geometric conditions (depending on p) leading to interpolation error estimates in the W1,p norm, for any p ϵ [1,∞), which can be regarded as a generalization of the regular decomposition property (RDP) condition introduced in [G. Acosta and R. G. Durán, SIAM J. Numer. Anal., 38 (2000), pp. 1073-1088] for p = 2 and new RDP conditions (NRDP) for p ≠ 2. We avoid the use of the reference family elements, which allows us to extend the results to a larger class of elements and to introduce the NRDP condition in a more unified way. As far as we know, the mesh condition presented in this paper is weaker than any other mesh conditions proposed in the literature for any p with 1 ≤ p≤∞. [ABSTRACT FROM AUTHOR]

In this paper, we perform an a posteriori error analysis of a multiscale operator decomposition finite element method for the solution of a system of coupled elliptic problems. The goal is to compute accurate error estimates that account for the effects arising from multiscale discretization via operator decomposition. Our approach to error estimation is based on a well-known a posteriori analysis involving variational analysis, residuals, and the generalized Green's function. Our method utilizes adjoint problems to deal with several new features arising from the multiscale operator decomposition. In part I of this paper, we focus on the propagation of errors arising from the solution of one component to another and the transfer of information between different representations of solution components. We also devise an adaptive discretization strategy based on the error estimates that specifically controls the effects arising from operator decomposition. In part II of this paper, we address issues related to the iterative solution of a fully coupled nonlinear system. [ABSTRACT FROM AUTHOR]