Convergence of relaxation schemes to the equations of elastodynamics

We study the effect of approximation matrices to semi-discrete relaxation schemes for the equations of one-dimensional elastodynamics. We consider a semi-discrete relaxation scheme and establish convergence using the theory of compensated compactness. Then we study the convergence of an associated relaxation-diffusion system, inspired by the scheme. Numerical comparisons of fully-discrete schemes are carried out.

     

Good and viscosity solutions of fully nonlinear elliptic equations

We introduce the notion of a ``good" solution of a fully nonlinear uniformly elliptic equation. It is proven that ``good" solutions are equivalent to -viscosity solutions of such equations. The main contribution of the paper is an explicit construction of elliptic equations with strong solutions that approximate any given fully nonlinear uniformly elliptic equation and its -viscosity solution. The results also extend some results about ``good" solutions of linear equations.

     

Convergence of the shifted algorithm for unitary Hessenberg matrices

This paper shows that for unitary Hessenberg matrices the algorithm, with (an exceptional initial-value modification of) the Wilkinson shift, gives global convergence; moreover, the asymptotic rate of convergence is at least cubic, higher than that which can be shown to be quadratic only for Hermitian tridiagonal matrices, under no further assumption. A general mixed shift strategy with global convergence and cubic rates is also presented.

     

An analysis of nonconforming multi-grid methods, leading to an improved method for the Morley element

We recall and slightly refine the convergence theory for nonconforming multi-grid methods for symmetric positive definite problems developed by Bramble, Pasciak and Xu. We derive new results to verify the regularity and approximation assumption, and the assumption on the smoother. From the analysis it will appear that most efficient multi-grid methods can be expected for fully regular problems, and for prolongations for which the energy norm of the iterated prolongations is uniformly bounded.

Guided by these observations, we develop a new multi-grid method for the biharmonic equation discretized with Morley finite elements, or equivalently, for the Stokes equations discretized with the -nonconforming pair. Numerical results show that the new method is superior to standard ones.

     

Discrete compactness and the approximation of Maxwell's equations in

We analyze the use of edge finite element methods to approximate Maxwell's equations in a bounded cavity. Using the theory of collectively compact operators, we prove -convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are based on the discrete compactness property of edge element due to Kikuchi. We extend the original work of Kikuchi by proving that edge elements of all orders possess this property.

     

Rate of convergence of finite difference approximations for degenerate ordinary differential equations

In this paper we study finite difference approximations for the following linear stationary convection-diffusion equations:

where is allowed to be degenerate. We first propose a new weighted finite difference scheme, motivated by approximating the diffusion process associated with the equation in the strong sense. We show that, under certain conditions, this scheme converges with the first order rate and that such a rate is sharp. To the best of our knowledge, this is the first sharp result in the literature. Moreover, by using the connection between our scheme and the standard upwind finite difference scheme, we get the rate of convergence of the latter, which is also new.

     

Homogeneous solutions to fully nonlinear elliptic equations

We classify homogeneous degree solutions to fully nonlinear elliptic equations.

     

Conformally invariant Monge-Ampère equations: Global solutions

In this paper we will examine a class of fully nonlinear partial differential equations which are invariant under the conformal group . These equations are elliptic and variational. Using this structure and the conformal invariance, we will prove a global uniqueness theorem for solutions in with a quadratic growth condition at infinity.

     

Improving the convergence of non-interior point algorithms for nonlinear complementarity problems

Recently, based upon the Chen-Harker-Kanzow-Smale smoothing function and the trajectory and the neighbourhood techniques, Hotta and Yoshise proposed a noninterior point algorithm for solving the nonlinear complementarity problem. Their algorithm is globally convergent under a relatively mild condition. In this paper, we modify their algorithm and combine it with the superlinear convergence theory for nonlinear equations. We provide a globally linearly convergent result for a slightly updated version of the Hotta-Yoshise algorithm and show that a further modified Hotta-Yoshise algorithm is globally and superlinearly convergent, with a convergence -order , under suitable conditions, where is an additional parameter.

     

On the asymptotic linearization of acoustic waves

The initial value problem of a certain generalization of the nonlinear, dispersive wave equations with dissipation is rigorously studied. The solutions of the equations can be found exactly up to in certain norms. The essential use is made of the fact that this equation is asymptotically linearizable to i.e., the equations can be mapped to an equation which differs from a linearizable equation only in terms which are of An application of the equations to unidirectional small amplitude acoustic waves is discussed. The general methodology used here can also be applied to other asymptotically linearizable equations.

     

Directional Newton methods in variables

Directional Newton methods for functions of variables are shown to converge, under standard assumptions, to a solution of . The rate of convergence is quadratic, for near-gradient directions, and directions along components of the gradient of with maximal modulus. These methods are applied to solving systems of equations without inversion of the Jacobian matrix.

     

Strong converse inequality for Poisson sums

The rate of convergence of Poisson sums and their combinations are shown to be equivalent to appropriate -functionals.

     

Boundary element monotone iteration scheme for semilinear elliptic partial differential equations

The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the Aubin-Nitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundary-element iterates with respect to the , , Sobolev space norms. Such estimates are of optimal order. Furthermore, as a peculiarity, we show that for the nonlinearities that are of separable type, ``higher than optimal order' error estimates can be obtained with respect to the mesh parameter . Several examples of semilinear elliptic partial differential equations featuring different situations of existence/nonexistence, uniqueness/multiplicity and stability are discussed, computed, and the graphics of their numerical solutions are illustrated.

     

HP a-priori error estimates for a non-dissipative spectral discontinuous Galerkin method to solve the Maxwell equations in the time domain

In this paper, we present the -convergence analysis of a non-dissipative high-order discontinuous Galerkin method on unstructured hexahedral meshes using a mass-lumping technique to solve the time-dependent Maxwell equations. In particular, we underline the spectral convergence of the method (in the sense that when the solutions and the data are very smooth, the discretization is of unlimited order). Moreover, we see that the choice of a non-standard approximate space (for a discontinuous formulation) with the absence of dissipation can imply a loss of spatial convergence. Finally we present a numerical result which seems to confirm this property.

     

Convergence study of the Chorin-Marsden formula

Using the fundamental solution of the heat equation, we give an expression of the solutions to two-dimensional initial-boundary value problems of the Navier-Stokes equations, where the vorticity is expressed in terms of a Poisson integral, a Newtonian potential, and a single layer potential. The density of the single layer potential is the solution to an integral equation of Volterra type along the boundary. We prove there is a unique solution to the integral equation. One fractional time step approximation is given, based on this expression. Error estimates are obtained for linear and nonlinear problems. The order of convergence is for the Navier-Stokes equations. The result is in the direction of justifying the Chorin-Marsden formula for vortex methods. It is shown that the density of the vortex sheet is twice the tangential velocity for the half plane, while in general the density differs from it by one additional term.

     

On the behavior of weak convergence under nonlinearities and applications

This paper provides a sufficient condition to guarantee the stability of weak limits under nonlinear operators acting on vector-valued Lebesgue spaces. This nonlinear framework places the weak convergence in perspective. Such an approach allows short and insightful proofs of important results in Functional Analysis such as: weak convergence in implies strong convergence in for all , weak convergence in vs. strong convergence in and the Brezis-Lieb theorem. The final goal is to use this framework as a strategy to grapple with a nonlinear weak spectral problem on .

     

A three-curves theorem for viscosity subsolutions of parabolic equations

We prove a three-curves theorem for viscosity subsolutions of fully nonlinear uniformly parabolic equations .

     

On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions

We present nonnegativity-preserving finite element schemes for a general class of thin film equations in multiple space dimensions. The equations are fourth order degenerate parabolic, and may contain singular terms of second order which are to model van der Waals interactions. A subtle discretization of the arising nonlinearities allows us to prove discrete counterparts of the essential estimates found in the continuous setting. By use of the entropy estimate, strong convergence results for discrete solutions are obtained. In particular, the limit of discrete fluxes will be identified with the flux in the continuous setting. As a by-product, first results on existence and positivity almost everywhere of solutions to equations with singular lower order terms can be established in the continuous setting.

     

The approximate inverse in action II: convergence and stability

The approximate inverse is a scheme for constructing stable inversion formulas for operator equations. Originally, it is defined on -spaces. In the present article we extend the concept of approximate inverse to more general settings which allow us to investigate the discrete version of the approximate inverse which actually underlies numerical computations. Indeed, we show convergence if the discretization parameter tends to zero. Further, we prove stability, that is, we show the regularization property. Finally we apply the results to the filtered backprojection algorithm in 2D-tomography to obtain convergence rates.

     

Log-log convexity and backward uniqueness

We study backward uniqueness properties for equations of the form

Under mild regularity assumptions on and , it is shown that implies for . The argument is based on -log and log-log convexity. The results apply to mildly nonlinear parabolic equations and systems with rough coefficients and the 2D Navier-Stokes system.