Deriving a discrete approximate solution for a nonlinear dynamic system of two-phase soil media

A discrete approximate generalized solution is derived for a nonlinear differential model of the dynamics of two-phase soil media and its convergence is estimated for the corresponding generalized solution in the space W ₂ ¹ (Ω). Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 69–80, July–August 2009     

Convergence of an upwind control-volume mixed finite element method for convection–diffusion problems

Summary We consider a upwind control volume mixed finite element method for convection–diffusion problem on rectangular grids. These methods use the lowest order Raviart–Thomas mixed finite element space as the trial functional space and associate control-volumes, or covolumes, with the vector variable as well as the scalar variable. Chou et al. [6] established a one-half order convergence in discrete L ²-norms. In this paper, we establish a first order convergence for both the vector variable as well as the scalar variable in discrete L ²-norms.      

Numerical analysis of electrokinetic transport in micro-nanofluidic interconnect preconcentrator in hydrodynamic flow

The phenomenon of enrichment of charged analytes due to the presence of an electric field barrier at the micro-nanofluidic interconnect can be harnessed to enhance sensitivity and limit-of-detection in sensor instruments. We present a numerical analysis framework to investigate two critical electrokinetic phenomena underlying the experimental observation in Plecis et al. (Micro Total Analysis Systems, pp 1038–1041, 2005b): (1) ion transport of background electrolytes (BGE) and (2) enrichment of analytes in the micro-nanofluidic devices that operate under hydrodynamic flow. The analysis is based on the full, coupled solution of the Poisson–Nernst–Planck (PNP) and Naviér–Stokes equations, and the results are validated against analytical models of simple canonical geometry. Parametric simulation is performed to capture the critical effects of pressure head and BGE ion concentration on the electrokinetics and ion transport. Key findings obtained from the numerical analysis indicate that the hydrodynamic flow and overlapped electrical double layer induce concentration–polarization at the interfaces; significant electric field barrier arising from the Donnan potential forms at the micro–nano interfaces; and streaming potential and overall potential are effectively established across the micro-nanofluidic device. The simulation to examine analyte enrichment and its dependence on the hydrodynamic flow and analyte properties, demonstrates that order-of-magnitude enrichment can be achieved using properly configured hydrodynamic flow. The results can be used to guide practical design and operational protocol development of novel micro-nanofluidic interconnect-based analyte preconcentrators.     

The Tau Method as an analytic tool in the discussion of equivalence results across numerical methods

A Tau Method approximate solution of a given differential equation defined on a compact [a, b] is obtained by adding to the right hand side of the equation a specific minimal polynomial perturbation termH _n(x), which plays the role of a representation of zero in [a,b] by elements of a given subspace of polynomials. Neither discretization nor orthogonality are involved in this process of approximation. However, there are interesting relations between the Tau Method and approximation methods based on the former techniques. In this paper we use equivalence results for collocation and the Tau Method, contributed recently by the authors together with classical results in the literature, to identify precisely the perturbation termH(x) which would generate a Tau Method approximate solution, identical to that generated by some specific discrete methods over a given mesh Π ∈ [a, b]. Finally, we discuss a technique which solves the inverse problem, that is, to find adiscrete perturbed Runge-Kutta scheme which would simulate a prescribed Tau Method. We have chosen, as an example, a Tau Method which recovers the same approximation as an orthogonal expansion method. In this way we close the diagram defined by finite difference methods, collocation schemes, spectral techniques and the Tau Method through a systematic use of the latter as an analytical tool.     

Computational investigation of elliptic systems with discontinuous coefficients

The Dirichlet problem is considered for a linear elliptic system of the second order with discontinuous coefficients in a rectangle Ω. For this problem, a difference scheme of the order of accuracy is constructed in a net norm W ₂ ¹ (ω), provided that the solution of the original problem belongs to the space W ₂ ^k (Ω_α), α= 1,2, k=2,3, in each of the subdomains into which the rectangle is divided by the curve of discontinuity of coefficients. This work was supported by the State Foundation for Basic Research. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 157–161, July–August, 2000.     

Conforming spectral methods for Poisson problems in cuboidal domains

A Chebyshev collocation strategy is introduced for the subdivision of cuboids into cuboidal subdomains (elements). These elements are conforming, which means that the approximation to the solution isC ⁰ continuous at all points across their interfaces.     

A nonconforming finite element method¶with anisotropic mesh grading¶for the incompressible Navier–Stokes equations¶in domains with edges

Any solution of the incompressible Navier–Stokes equations in three-dimensional domains with edges has anisotropic singular behaviour which is treated numerically by using anisotropic finite element meshes. The velocity is approximated by Crouzeix–Raviart (nonconforming 𝒫₁) elements and the pressure by piecewise constants. This method is stable for general meshes since the inf-sup condition is satisfied without minimal or maximal angle condition. The existence of solutions to the discrete problems follows. Consistency error estimates for the divergence equation are obtained for anisotropic tensor product meshes. As applications, the consistency error estimate for the Navier–Stokes solution and some discrete Sobolev inequalities are derived on such meshes. These last results provide optimal error estimates in the uniqueness case by the use of appropriately refined anisotropic tensor product meshes, namely, if N _e is the number of elements of the mesh, we prove that the optimal order of convergence h∼ N _e ^{− 1/3}. Received:July 2001 / Accepted: July 2002     

A rotated nonconforming rectangular mixed element for elasticity

We present a low-order nonconforming mixed element for plane elasticity on rectangular meshes. The 3-dimensional space of rigid body motions is used to approximate the displacement and a 16-dimensional space is used to discretize the space of symmetric tensors. This element may be viewed as the rectangular analogue of the nonconforming Arnold-Winther element and is related to a discrete version of the elasticity differential complex with a nonconforming H ² element related to the rotated Q ₁ element.      

Polynomial time-marching for nonperiodic boundary value problems

A polynomial interpolation time-marching technique can efficiently provide balanced spectral accuracy in both the space and time dimensions for some PDEs. The Newton-form interpolation based on Fejér points has been successfully implemented to march the periodic Fourier pseudospectral solution in time. In this paper, this spectrally accurate time-stepping technique will be extended to solve some typical nonperiodic initial boundary value problems by the Chebyshev collocation spatial approximation. Both homogeneous Neumann and Dirichlet boundary conditions will be incorporated into the time-marching scheme. For the second order wave equation, besides more accurate timemarching, the new scheme numerically has anO(1/N ²) time step size limitation of stability, much larger thanO(1/N ⁴) stability limitation in conventional finitedifference time-stepping, Chebyshev space collocation methods.     

On the numerical solution of the eigenvalue problem of the laplace operator by a capacitance matrix method

The problem of finding several eigenfunctions and eigenvalues of the interior Dirichlet problem for Laplace's equation on arbitrary bounded plane regions is considered. Two fast algorithms are combined: an iterative Block Lanczos method and a capacitance matrix method. The capacitance matrix is generated and factored only once for a given problem. In each iteration of the Block Lanczos method, a discrete Helmholtz equation is solved twice on a rectangle at a cost of the order ofn ² log₂ n operations wheren is the number of mesh points across the rectangle in which the region is imbedded.     

Direct Numerical Simulation in a Lid-Driven Cubical Cavity at High Reynolds Number by a Chebyshev Spectral Method

Direct numerical simulation of the flow in a lid-driven cubical cavity has been carried out at high Reynolds numbers (based on the maximum velocity on the lid), between 1.2 10⁴ and 2.2 10⁴. An efficient Chebyshev spectral method has been implemented for the solution of the incompressible Navier–Stokes equations in a cubical domain. The Projection-Diffusion method [Leriche and Labrosse (2000, SIAM J. Sci. Comput. 22(4), 1386–1410), Leriche et al. (2005, J. Sci. Comput., in press)] allows to decouple the velocity and pressure computation in very efficient way and the simple geometry allows to use the fast diagonalisation method for inverting the elliptic operators at a low computational cost. The resolution used up to 5.0 million Chebyshev collocation nodes, which enable the detailed representation of all dynamically significant scales of motion. The mean and root-mean-square velocity statistics are briefly presented     

Splines in Higher Order TV Regularization

Splines play an important role as solutions of various interpolation and approximation problems that minimize special functionals in some smoothness spaces. In this paper, we show in a strictly discrete setting that splines of degree m−1 solve also a minimization problem with quadratic data term and m-th order total variation (TV) regularization term. In contrast to problems with quadratic regularization terms involving m-th order derivatives, the spline knots are not known in advance but depend on the input data and the regularization parameter λ. More precisely, the spline knots are determined by the contact points of the m–th discrete antiderivative of the solution with the tube of width 2λ around the m-th discrete antiderivative of the input data. We point out that the dual formulation of our minimization problem can be considered as support vector regression problem in the discrete counterpart of the Sobolev space W _2,0 ^m . From this point of view, the solution of our minimization problem has a sparse representation in terms of discrete fundamental splines.     

A Hierarchical 3-D Poisson Modified Fourier Solver by Domain Decomposition

We present a Domain Decomposition non-iterative solver for the Poisson equation in a 3-D rectangular box. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the non-homogeneous equation is first computed by a fast spectral method. This method is based on the application of the discrete Fourier transform accompanied by a subtraction technique. For high accuracy the subdomain boundary conditions must be compatible with the specified inhomogeneous right hand side at the edges of all the interfaces. In the following steps the partial solutions are hierarchically matched. At each step pairs of adjacent subdomains are merged into larger units. In this paper we present the matching algorithm for two boxes which is a basis of the domain decomposition scheme. The hierarchical approach is convenient for parallelization and minimizes the global communication. The algorithm requires O(N ³:log:N) operations, where N is the number of grid points in each direction.     

A General Reliable Quadratic Form: An Extension of Affine Arithmetic

In this article, a new extension of affine arithmetic is introduced. This technique is based on a quadratic form named general quadratic form. We focus here on the computation of reliable bounds of a function over a hypercube by using this new tool. Some properties of first quadratic functions and then polynomial ones are reported. In order to show the efficiency of such a method, ten polynomial global optimization problems are presented and solved by using an interval branch-and-bound based algorithm. The work of the first author was also supported by the Laboratoire de Mathématiques Appliquées CNRS–FRE 2570, Université de Pau et des Pays de l'Adour, France, and by the Laboratoire d'Electrotechnique et d'Electronique Industrielle CNRS–UMR5828, Group EM³, INPT–ENSEEIHT.     

Remarks on some mixed finite element schemes for Reissner–Mindlin plate model

Two different stabilization procedures for mixed finite element schemes for Reissner–Mindlin plate problems are introduced. They are based on a suitable modification of the discrete shear energy like that introduced when a partial selective reduced integration technique is used. Some numerical results will be presented in order to show the performance of these schemes with respect to the locking phenomenon. The dependence of the approximate solution on the stabilizing parameter is also analyzed. Received: October 2001 / Accepted: March 2002     

Dispersion Analysis of Discontinuous Galerkin Schemes Applied to Poincaré, Kelvin and Rossby Waves

A technique for analyzing dispersion properties of numerical schemes is proposed. The method is able to deal with both non dispersive or dispersive waves, i.e. waves for which the phase speed varies with wavenumber. It can be applied to unstructured grids and to finite domains with or without periodic boundary conditions. We consider the discrete version L of a linear differential operator ℒ. An eigenvalue analysis of L gives eigenfunctions and eigenvalues (l _i ,λ _i ). The spatially resolved modes are found out using a standard a posteriori error estimation procedure applied to eigenmodes. Resolved eigenfunctions l _i ’s are used to determine numerical wavenumbers k _i ’s. Eigenvalues’ imaginary parts are the wave frequencies ω _i and a discrete dispersion relation ω _i =f(k _i ) is constructed and compared with the exact dispersion relation of the continuous operator ℒ. Real parts of eigenvalues λ _i ’s allow to compute dissipation errors of the scheme for each given class of wave. The method is applied to the discontinuous Galerkin discretization of shallow water equations in a rotating framework with a variable Coriolis force. Such a model exhibits three families of dispersive waves, including the slow Rossby waves that are usually difficult to analyze. In this paper, we present dissipation and dispersion errors for Rossby, Poincaré and Kelvin waves. We exhibit the strong superconvergence of numerical wave numbers issued of discontinuous Galerkin discretizations for all families of waves. In particular, the theoretical superconvergent rates, demonstrated for a one dimensional linear transport equation, for dissipation and dispersion errors are obtained in this two dimensional model with a variable Coriolis parameter for the Kelvin and Poincaré waves.     

<Emphasis Type="Italic">hp</Emphasis>-Version <Emphasis Type="Italic">a priori</Emphasis> Error Analysis of Interior Penalty Discontinuous Galerkin Finite Element Approximations to the Biharmonic Equation

We consider the symmetric formulation of the interior penalty discontinuous Galerkin finite element method for the numerical solution of the biharmonic equation with Dirichlet boundary conditions in a bounded polyhedral domain in . For a shape-regular family of meshes consisting of parallelepipeds, we derive hp-version a priori bounds on the global error measured in the L² norm and in broken Sobolev norms. Using these, we obtain hp-version bounds on the error in linear functionals of the solution. The bounds are optimal with respect to the mesh size h and suboptimal with respect to the degree of the piecewise polynomial approximation p. The theoretical results are confirmed by numerical experiments, and some practical applications in Poisson–Kirchhoff thin plate theory are presented.     

On the div-curl lemma in a Galerkin setting

Given a sequence of Galerkin spaces X _h of square-integrable vector fields, we state necessary and sufficient conditions on X _h under which it is true that for any two sequences of vector fields u _h ,u _h ′∈X _h converging weakly in L² and such that u _h is discrete divergence free and curl u _h ′ is precompact in H⁻¹, the scalar product u _h ⋅u′ _h converges in the sense of distributions to the right limit. The conditions are related to super-approximation and discrete compactness results for mixed finite elements, and are satisfied for Nédélec’s edge elements. We also provide examples of sequences of discrete divergence free edge element vector fields converging weakly to 0 in L² but whose divergence is not precompact in H _loc ⁻¹.      

A Note on the Spectral Collocation Approximation of Some Differential Equations with Singular Source Terms

The solution of differential equations with singular source terms contains the local jump discontinuity in general and its spectral approximation is oscillatory due to the Gibbs phenomenon. To minimize the Gibbs oscillations near the local jump discontinuity and improve convergence, the regularization of the approximation is needed. In this note, a simple derivative of the discrete Heaviside function H _c (x) on the collocation points is used for the approximation of singular source terms δ(x−c) or δ ⁽ⁿ⁾(x−c) without any regularization. The direct projection of H _c (x) yields highly oscillatory approximations of δ(x−c) and δ ⁽ⁿ⁾(x−c). In this note, however, it is shown that the direct projection approach can yield a non-oscillatory approximation of the solution and the error can also decay uniformly for certain types of differential equations. For some differential equations, spectral accuracy is also recovered. This method is limited to certain types of equations but can be applied when the given equation has some nice properties. Numerical examples for elliptic and hyperbolic equations are provided. The current address: Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, USA.