IMA Journal of Numerical Analysis - recent issues

Compact embeddings of broken Sobolev spaces and applications

Buffa, A., Ortner, C. — Tue, 27 Oct 2009 03:48:04 PDT

In this paper, we present several extensions of theoretical tools for the analysis of discontinuous Galerkin (DG) method beyond the linear case. We define broken Sobolev spaces for Sobolev indices in [1, ), and we prove generalizations of many techniques of classical analysis in Sobolev spaces. Our targeted application is the convergence analysis for DG discretizations of energy minimization problems of the calculus of variations. Our main tool in this analysis is a theorem which permits the extraction of a ‘weakly’ converging subsequence of a family of discrete solutions and which shows that any ‘weak limit’ is a Sobolev function. As a second application, we compute the optimal embedding constants in broken Sobolev–Poincaré inequalities.

Discontinuous Galerkin approximations for Volterra integral equations of the first kind

Brunner, H., Davies, P. J., Duncan, D. B. — Tue, 27 Oct 2009 03:48:04 PDT

Motivated by the problem of developing accurate and stable time-stepping methods for the single-layer potential equation for acoustic scattering from a surface, we present new convergence results for piecewise polynomial discontinuous Galerkin (DG) approximations of a first-kind Volterra integral equation of convolution kernel type, where the kernel K is smooth and satisfies K(0) != 0. We show that an mth degree DG approximation exhibits global convergence of order m when m is odd and order m + 1 when m is even. There is local superconvergence of one order higher (i.e. order m + 1 when m is odd and m + 2 when m is even), but in the even order case, there is superconvergence only if the exact solution u of the equation satisfies u^{(m + 1)}(0) = 0. We also present numerical test results which show that these theoretical convergence rates are optimal.

From high oscillation to rapid approximation III: multivariate expansions

Iserles, A., Norsett, S. P. — Tue, 27 Oct 2009 03:48:04 PDT

In this paper, we expand upon the theme of modified Fourier expansions and extend the theory to a multivariate setting and to expansions in eigenfunctions of the Laplace–Neumann operator. We pay detailed attention to expansions in a d-dimensional cube and to an effective derivation of expansion coefficients there by means of quadratures of highly oscillatory integrals. Thus, we present asymptotic and Filon-type formulae for an effective derivation of expansion coefficients and discuss their design and relative advantages. Such methods are effective only for large indices; hence, we introduce and analyse alternative quadrature schemes that require a relatively modest number of additional function evaluations.

Convergence rates for adaptive finite elements

Gaspoz, F. D., Morin, P. — Tue, 27 Oct 2009 03:48:04 PDT

In this article, we prove that it is possible to construct, using newest vertex bisection, meshes that equidistribute the error in the H¹-norm whenever the function to be approximated can be decomposed as a sum of a regular part plus a singular part with singularities around a finite number of points. This decomposition is usual in regularity results of partial differential equations. As a consequence, the meshes turn out to be quasi-optimal, and convergence rates for adaptive finite-element methods using Lagrange finite elements of any polynomial degree are obtained.

Numerical approximation of corotational dumbbell models for dilute polymers

Barrett, J. W., Suli, E. — Tue, 27 Oct 2009 03:48:04 PDT

We construct a general family of Galerkin methods for the numerical approximation of weak solutions to a bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function satisfying a Fokker–Planck-type parabolic equation. We focus on finitely extensible nonlinear elastic-type dumbbell models. In the case of a corotational drag term, we perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero and show that a (sub)sequence of numerical solutions converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system.

Convolution of hp-functions on locally refined grids

Hackbusch, W. — Tue, 27 Oct 2009 03:48:04 PDT

Usually, the fast evaluation of a convolution integral requires that the functions f and g have a simple structure based on an equidistant grid in order to apply the fast Fourier transform. Here, we discuss the efficient performance of the convolution of hp-functions in certain locally refined grids. More precisely, the convolution result is projected into some given hp-space (Galerkin approximation). The overall cost is O(p²N log N), where N is the sum of the dimensions of the subspaces containing f, g and the resulting function, while p is the maximal polynomial degree.

A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions

Liu, F., Madden, N., Stynes, M., Zhou, A. — Tue, 27 Oct 2009 03:48:04 PDT

The linear reaction–diffusion problem – ²u + bu = f is considered on the unit square with homogeneous Dirichlet boundary conditions. Here is a small positive parameter and the problem is in general singularly perturbed. The numerical solution of this problem is analysed on a Shishkin mesh that has N intervals in each coordinate direction, using the Galerkin finite-element method with bilinear trial functions. The accuracy of this method, measured in the associated energy norm, is shown to be O(N^–2 + ^1/2N^–1 ln N). It is proved that a two-scale sparse grid method achieves the same order of accuracy while reducing the number of degrees of freedom from O(N²) to O(N^3/2). These results are then generalized to systems of reaction–diffusion equations.

Enhancing eigenvalue approximation by gradient recovery on adaptive meshes

Wu, H., Zhang, Z. — Tue, 27 Oct 2009 03:48:04 PDT

We utilize the recovered gradient by the polynomial-preserving recovery to enhance the eigenvalue approximation of the Laplace operator under adaptive meshes. Superconvergence rate is established and numerical tests on benchmark problems support our theoretical findings.

Gauss-Hermite wave packet dynamics: convergence of the spectral and pseudo-spectral approximation

Faou, E., Gradinaru, V. — Tue, 27 Oct 2009 03:48:05 PDT

The time-dependent linear Schrödinger equation for nuclei on the whole space is semidiscretized using Hermite and Gauss–Hermite basis functions. These are well suited, on the one hand, for the conservation properties of the numerical solution and, on the other hand, for their remarkable approximation properties. We investigate theoretically and numerically the convergence of the spectral and pseudo-spectral Gauss–Hermite semidiscretization schemes.

Some numerical methods for second-kind Fredholm integral equations on the real semiaxis

Mastroianni, G., Milovanovic, G. V. — Tue, 27 Oct 2009 03:48:05 PDT

In this paper we introduce some numerical methods for solving Fredholm integral equations of the second kind on the real semiaxis and prove that the proposed procedures are stable and convergent. Error estimates and numerical tests are also included.

The QR algorithm: 50 years later its genesis by John Francis and Vera Kublanovskaya and subsequent developments

Golub, G., Uhlig, F. — Thu, 02 Jul 2009 20:30:32 PDT

Fifty years after the invention of the QR algorithm by John Francis and Vera Kublanovskaya we reconstruct the ideas and the influences that led to its genesis from the originators’ own recollections and their sources and give an account of some of its subsequent developments.

Error estimates for Gauss-Turan quadratures and their Kronrod extensions

Milovanovic, G. V., Spalevic, M. M., Pranic, M. S. — Thu, 02 Jul 2009 20:30:32 PDT

We study the kernel K_n_{, s}(z) of the remainder term R_n_{, s}(f) of Gauss–Turán–Kronrod quadrature rules with respect to one of the generalized Chebyshev weight functions for analytic functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L-error bounds of Gauss–Turán–Kronrod quadratures. Following Kronrod, using the modulus of the difference of Gauss–Turán quadratures and their Kronrod extensions, we derive new error estimates for Gauss–Turán quadratures and compare them with the effective L¹-error bounds derived in Milovanovic & Spalevic (2005, BIT, 45, 117–136).

Alternate slice-based substructuring in three dimensions

Mihai, L. A., Craig, A. W. — Thu, 02 Jul 2009 20:30:32 PDT

The slice-based substructuring methods introduced in this paper are Schur complement solvers for the decomposition of a three-dimensional domain into multiple disjoint subdomains with interior crosspoints. The subdomains are assembled into nonoverlapping slices such that the edges of each slice lie on the boundary of the given domain and the union of the faces between slices contains all of the interior vertices. For the subproblems corresponding to the various faces, a direct fast Poisson solver is used. Scalability is achieved in two stages where the slices change such that the faces between slices at one stage are orthogonal to the faces between slices at the other. The two stages guarantee a good rate of convergence of the resulting preconditioned iterative procedure, which is optimal with respect to the partitioning parameters.

A convergent finite-difference method for a nonlinear variational wave equation

Holden, H, Karlsen, K. H., Risebro, N. H. — Thu, 02 Jul 2009 20:30:32 PDT

We establish rigorously convergence of a semidiscrete upwind scheme for the nonlinear variational wave equation u_tt – c(u)(c(u)u_x)_x = 0 with u|_t₌₀ = u₀ and u_t|_t=0 = v₀. Introducing Riemann invariants R = u_t + cu_x and S = u_t – cu_x, the variational wave equation is equivalent to R_t – cR_x (R² – S²) and S_t + cS_x = – (R² – S²) with = c'/(4c). An upwind scheme is defined for this system. We assume that the speed c is positive, increasing and both c and its derivative are bounded away from zero and that R|_t₌₀, S|_t₌₀ L¹(R) L³(R) are nonpositive. The numerical scheme is illustrated on several examples.

Discontinuous Galerkin methods for the biharmonic problem

Georgoulis, E. H., Houston, P. — Thu, 02 Jul 2009 20:30:32 PDT

This work is concerned with the design and analysis of hp-version discontinuous Galerkin (DG) finite element methods for boundary-value problems involving the biharmonic operator. The first part extends the unified approach of Arnold et al. (2001/2002, SIAM J. Numer. Anal., 39, 1749-–1779) developed for the Poisson problem, to the design of DG methods via an appropriate choice of numerical flux functions for fourth-order problems; as an example, we retrieve the interior penalty DG method developed by Süli & Mozolevski (2007, Comput. Methods Appl. Mech. Eng., 196, 1851-–1863). The second part of this work is concerned with a new a priori error analysis of the hp-version interior penalty DG method, when the error is measured in terms of both the energy norm and the L²-norm, as well as certain linear functionals of the solution, for elemental polynomial degrees p ≥ 2. Also, provided that the solution is piecewise analytic in an open neighbourhood of each element, exponential convergence is also proved for the p-version of the DG method. The sharpness of the theoretical developments is illustrated by numerical experiments.

Modulated Fourier expansions and heterogeneous multiscale methods

Sanz-Serna, J. M. — Thu, 02 Jul 2009 20:30:32 PDT

We show that, for highly oscillatory ordinary differential equation problems, the modulated Fourier expansion approach can be advantageously used to understand and analyse the heterogeneous multiscale methods introduced by E, Engquist and their co-workers.

Continuous and discrete parabolic operators and their qualitative properties

Farago, I., Horvath, R. — Thu, 02 Jul 2009 20:30:32 PDT

The basic requirement of numerical methods is convergence. However, from the practical point of view, it is generally not sufficient to construct convergent numerical methods for the solutions of partial differential equations. The qualitative adequateness of the methods is also an issue. The numerical solutions should mirror the characteristic properties of the original physical process that is modelled by the differential equation. In this paper, we give three important qualitative properties of parabolic partial differential equations: the maximum–minimum principle and its different versions, the non-negativity preservation and the maximum norm contractivity. The investigation of these properties is motivated by different physical principles. We formulate the analogues of the properties for general discrete operators and we analyse the conditions and the relations between the properties for both the continuous and the discrete operators. The approximation properties of the discrete operators are also analysed. The results of the paper are applied to the finite-difference solution methods of parabolic initial boundary-value problems.

Nystrom method for systems of integral equations on the real semiaxis

De Bonis, M. C., Mastroianni, G. — Thu, 02 Jul 2009 20:30:32 PDT

In this paper, the authors introduce a Nyström method for solving systems of Fredholm integral equations on the real semiaxis. They prove that the method is stable and convergent. Moreover, they show some numerical tests that confirm the error estimates. Finally, they discuss a specific application to an inverse scattering problem for the Schrödinger equation.

Numerical analysis of the TV regularization and H-1 fidelity model for decomposing an image into cartoon plus texture

Elliott, C. M., Smitheman, S. A. — Thu, 02 Jul 2009 20:30:32 PDT

The Osher–Solé–Vese (OSV) model, which is the gradient flow of an energy consisting of the total variation functional plus an H^–1 fidelity term, is studied. In this paper, we build on the analysis of the OSV model which we gave in Elliott & Smitheman (2007, Comm. Pure Appl. Anal., in press). We introduce backward Euler finite-element approximations to a regularized version of the OSV initial boundary-value problem (IBVP) and to a weak formulation of the original problem. Well-posedness and unconditional Lyapunov stability of these fully discrete schemes are proved. Convergence results as the spatial mesh parameter, the time step size and the regularization parameter tend to 0 are proved. Rates of convergence as the time step size and the regularization parameter tend to 0 are found. The existence, uniqueness and Lyapunov stability of a solution to a linearly implicit finite-element approximation to the regularized version of the OSV IBVP are also proved.

Attractors of set-valued partial differential equations under discretization

Kloeden, P. E., Valero, J. — Thu, 02 Jul 2009 20:30:32 PDT

The approximation of the global attractor of a dissipative set-valued reaction–diffusion equation is investigated when a Galerkin approximation is used to obtain a finite-dimensional inclusion equation, to which the linear implicit Euler scheme is then applied. The existence and upper semicontinuous convergence of the various attractors with decreasing time step and increasing dimension are established. The equivalence of the attractors with those of the corresponding convexified systems is also shown.

Fitted mesh numerical methods for singularly perturbed elliptic problems with mixed derivatives

Dunne, R. K., O'Riordan, E., Shishkin, G. I. — Thu, 02 Jul 2009 20:30:32 PDT

A class of singularly perturbed convection–diffusion problems is considered which contain a mixed derivative term. We consider the case when exponential boundary layers are present in the solutions of problems from this class. Under appropriate assumptions on the data of the problem, we construct a decomposition of the solution into regular and layer components. We then introduce a numerical method on a piecewise-uniform fitted mesh. Excluding a neighbourhood of one of the corners, it is shown that in the perturbed case (i.e. when the perturbation parameter is small relative to the inverse of the number of mesh intervals in both coordinate directions), the approximations generated by the method converge uniformly with respect to the singular perturbation parameter. Finally, numerical examples are presented that illustrate the theoretical result.

Interpolation in special orthogonal groups

Shingel, T. — Thu, 02 Jul 2009 20:30:33 PDT

In this paper, we propose a scheme to generate interpolating curves in Lie groups, focusing on a special orthogonal group SO(n) due to its practical importance. Our technique is based on the exponential representation of the elements of the group, which allows as to transfer the problem to the corresponding Lie algebra (n) and benefit from the linearity of this space. Due to the exponential representation, we can obtain a high degree of smoothness of an interpolating curve. The underlying problem is challenging because the standard SO(n) -> (n) map is multivalued.

Local convergence of Newton's method in Banach space from the viewpoint of the majorant principle

Ferreira, O. P. — Thu, 02 Jul 2009 20:30:33 PDT

A local convergence analysis of Newton's method for solving nonlinear equations, based on Kantorovich's majorant principle, is presented in this paper. This analysis provides a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allows us to obtain the optimal convergence radius, the biggest range for the uniqueness of the solution, and to unify some previous and unrelated results.

Smoothness of interpolatory multivariate subdivision in Lie groups

Grohs, P. — Thu, 02 Jul 2009 20:30:33 PDT

Nonlinear subdivision schemes that operate on manifolds are of use whenever manifold-valued data have to be processed in a multiscale fashion. This paper considers the case where the manifold is a Lie group and the nonlinear subdivision schemes are derived from linear interpolatory ones by the so-called log–exp analogy. The main result of the paper is that a multivariate interpolatory Lie-group-valued subdivision scheme derived from a linear scheme is at least as smooth as the linear scheme, where smoothness is understood in terms of Hölder exponents.

A symmetric collocation method with fast evaluation

Johnson, M. J. — Thu, 02 Jul 2009 20:30:33 PDT

Symmetric collocation, which can be used to numerically solve linear partial differential equations, is a natural generalization of the well-established scattered data interpolation method known as radial basis function interpolation. As with radial basis function interpolation, a major shortcoming of symmetric collocation is the high cost, in terms of floating-point operations, of evaluating the obtained function. When solving a linear partial differential equation, one usually has some freedom in choosing the collocation points. We explain how this freedom can be exploited to allow the fast evaluation of the obtained function provided the basic function is chosen as a tensor product of compactly supported piecewise polynomials. Our proposed fast evaluation method, which is exact in exact arithmetic, is initially designed and analysed in the univariate case. The multivariate case is then reduced, recursively, to multiple univariate evaluations. Along with the theoretical development of the method, we report the results of selected numerical experiments which help to clarify expectations.

The LBB condition in fractional Sobolev spaces and applications

Guermond, J.-L. — Thu, 02 Jul 2009 20:30:33 PDT

The present work focuses on the approximation of the stationary Stokes equations by means of finite-element-like Galerkin methods. It is shown that, provided the velocity space and the pressure space are compatible in some sense, a Ladyzhenskaya–Babuska–Brezzi condition holds in the fractional Sobolev spaces H^s(), s [0, 1]. This result is illustrated in two applications.

Remarks on the implementation of the fast marching method

Rasch, C., Satzger, T. — Thu, 02 Jul 2009 20:30:33 PDT

The fast marching algorithm computes an approximate solution to the eikonal equation in time, where the factor log N is due to the administration of a priority queue. Recently, Yatziv et al. (2006 J. Comput. Phys., 212, 393–399) have suggested using an untidy priority queue, reducing the overall complexity to at the price of a small error in the computed solution. In this paper we give an explicit estimate of the error introduced, which is based on a discrete comparison principle. This estimate implies, in particular, that the choice of an accuracy level that is independent of the speed function F results in the complexity bound being . A numerical experiment illustrates this robustness problem for large ratios F max /F min.

A derivative-free nonmonotone line search and its application to the spectral residual method

Cheng, W., Li, D.-H. — Thu, 02 Jul 2009 20:30:33 PDT

In this paper we propose a derivative-free nonmonotone line search for solving large-scale nonlinear systems of equations. Under appropriate conditions, we show that the spectral residual method with this line search is globally convergent. We also present some numerical experiments. The results show that the spectral residual method with the new nonmonotone line search is promising.

A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity

Ern, A., Stephansen, A. F., Zunino, P. — Fri, 20 Mar 2009 12:17:51 PDT

We propose and analyse a symmetric weighted interior penalty method to approximate in a discontinuous Galerkin framework advection–diffusion equations with anisotropic and discontinuous diffusivity. The originality of the method consists in the use of diffusivity-dependent weighted averages to better cope with locally small diffusivity (or equivalently with locally high Péclet numbers) on fitted meshes. The analysis yields convergence results for the natural energy norm that are optimal with respect to mesh size and robust with respect to diffusivity. The convergence results for the advective derivative are optimal with respect to mesh size and robust for isotropic diffusivity, as well as for anisotropic diffusivity if the cell Péclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large enough. Numerical results are presented to illustrate the performance of the proposed scheme.

On the norm of the hyperinterpolation operator on the unit disc and its use for the solution of the nonlinear Poisson equation

Hansen, O., Atkinson, K., Chien, D. — Fri, 20 Mar 2009 12:17:51 PDT

In this article, we study the properties of the hyperinterpolation operator on the unit disc D in R². We show how hyperinterpolation can be used in connection with the Kumar–Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disc (discrete Galerkin method). A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results prove the convergence of the discrete Galerkin method in the maximum norm if the solution of the Poisson equation is in the class C^1, (D), > 0. Finally, we present numerical examples which show that the discrete Galerkin method converges faster than O(n^–k), for every k N if the solution of the nonlinear Poisson equation is in C(D).

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Burman, E., Guzman, J., Leykekhman, D. — Fri, 20 Mar 2009 12:17:51 PDT

In this paper, we analyse local properties of the continuous interior penalty (CIP) method for a model convection-dominated singularly perturbed convection–diffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain of interest. This result shows thats locally, the CIP method is comparable to the streamline-diffusion or the discontinuous Galerkin methods.

The spectral gradient method for unconstrained optimal control problems

Ardenghi, J. I., Gibelli, T. I., Maciel, M. C. — Fri, 20 Mar 2009 12:17:51 PDT

Optimal control problems and their discretized form can be viewed as optimization problems. Kelley and Sachs have already solved the discretized problem by using quasi-Newton methods. In this contribution, the problem is solved by a low-cost algorithm, the spectral gradient method, which is suitable for large-scale problems. The convergence behaviour of the method to finite-dimensional approximation is analysed. Numerical examples are given and the reported results show the good performance of the algorithm when it is applied to large optimal control problems.

A Lions non-overlapping domain decomposition method for domains with an arbitrary interface

Lui, S. H. — Fri, 20 Mar 2009 12:17:51 PDT

Lions' non-overlapping domain decomposition method for the solution of elliptic partial differential equations has been analysed extensively by many authors. There have been numerous works on the convergence of the iterative method as well as variations of it. In the present work, we analyse several formulations of Lions' method. For two of these, we show that the spectral radius of the operator in the fixed-point iteration for the interface boundary function is bounded above by 1 – O(h^1/2) when the optimal value (O(h^–1/2)) of the parameter in the Robin boundary condition along the artificial interface is used. While this result is already known for rectangular domains with a straight interface, our analysis is valid for essentially arbitrary geometry. The method of Guo and Hou, which is Lions' method with relaxation, has a simple proof of convergence using our method of analysis. Similarly, we provide a simple proof of convergence of the scheme of Deng.

Block-diagonal preconditioning for spectral stochastic finite-element systems

Powell, C. E., Elman, H. C. — Fri, 20 Mar 2009 12:17:51 PDT

Deterministic models of fluid flow and the transport of chemicals in flows in heterogeneous porous media incorporate partial differential equations (PDEs) whose material parameters are assumed to be known exactly. To tackle more realistic stochastic flow problems, it is fitting to represent the permeability coefficients as random fields with prescribed statistics. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions are averaged to obtain statistical properties of the solution variables. Alternatively, so-called stochastic finite-element methods (SFEMs) discretize the probabilistic dimension of the PDE directly leading to a single structured linear system. The latter approach is becoming extremely popular but its computational cost is still perceived to be problematic as this system is orders of magnitude larger than for the corresponding deterministic problem. A simple block-diagonal preconditioning strategy incorporating only the mean component of the random field coefficient and based on incomplete factorizations has been employed in the literature and observed to be robust, for problems of moderate variance, but without theoretical analysis. We solve the stochastic Darcy flow problem in primal formulation via the spectral SFEM and focus on its efficient iterative solution. To achieve optimal computational complexity, we base our block-diagonal preconditioner on algebraic multigrid. In addition, we provide new theoretical eigenvalue bounds for the preconditioned system matrix. By highlighting the dependence of these bounds on all the SFEM parameters, we illustrate, in particular, why enriching the stochastic approximation space leads to indefinite system matrices when unbounded random variables are employed.

Analysis of a mixed finite-volume discretization of fourth-order equations on general surfaces

Du, Q., Ju, L., Tian, L. — Fri, 20 Mar 2009 12:17:51 PDT

In this paper, we study a finite-volume method for the numerical solution of a model fourth-order partial differential equation defined on a smooth surface. The discretization is done via a surface mesh consisting of piecewise planar triangles and its dual surface polygonal tessellation. We provide an error estimate for the approximate solution under the H¹-norm on general regular meshes. Numerical experiments are carried out on various sample surfaces to verify the theoretical results. In addition, when the underlying mesh is constructed by the so-called constrained centroidal Voronoi meshes, we propose a numerically demonstrated superconvergent scheme to compute gradients more accurately.

Inf-sup stable finite-element pairs based on dual meshes and bases for nearly incompressible elasticity

Lamichhane, B. P. — Fri, 20 Mar 2009 12:17:51 PDT

We consider finite-element methods based on simplices to solve the problem of nearly incompressible elasticity. Two different approaches based, respectively, on dual meshes and dual bases are presented, where in both approaches pressure is discontinuous and can be statically condensed out from the system. These novel approaches lead to displacement-based low-order finite-element methods for nearly incompressible elasticity based on rigorous mathematical framework. Numerical results are provided to demonstrate the efficiency of the approach.

Stochastic variational integrators

Bou-Rabee, N., Owhadi, H. — Fri, 20 Mar 2009 12:17:51 PDT

This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds, akin to the Ornstein–Uhlenbeck theory of Brownian motion in a force field. The main result is to derive governing SDEs for such systems from a critical point of a stochastic action. Using this result, the paper derives Langevin-type equations for constrained mechanical systems and implements a stochastic analogue of Lagrangian reduction. These are easy consequences of the fact that the stochastic action is intrinsically defined. Stochastic variational integrators (SVIs) are developed using a discrete variational principle. The paper shows that the discrete flow of an SVI is almost surely symplectic and in the presence of symmetry almost surely momentum-map preserving. A first-order mean-squared convergent SVI for mechanical systems on Lie groups is introduced. As an application of the theory, SVIs are exhibited for multiple, randomly forced and torqued rigid bodies interacting via a potential.

Stability analysis of general linear methods for the nonautonomous pantograph equation

Huang, C. — Fri, 20 Mar 2009 12:17:51 PDT

This paper is concerned with the study of the stability of general linear methods for the nonautonomous pantograph equation. Linear and nonlinear problems are considered separately. We derive the asymptotic stability of numerical methods with strict stability at infinity for neutral equations. Also, we obtain some bounds for the error growth for algebraically stable methods applied to non-neutral equations.