Orthogonal spline collocation methods for biharmonic problems

Orthogonal spline collocation methods are formulated and analyzed for the solution of certain biharmonic problems in the unit square. Particular attention is given to the Dirichlet biharmonic problem which is solved using capacitance matrix techniques. Received November 11, 1996     

Superconvergence of the orthogonal spline collocation solution of Poisson's equation

Superconvergence phenomena have been observed numerically in the piecewise Hermite bicubic orthogonal spline collocation solution of Poisson's equation on a rectangle. The purpose of this article is to demonstrate theoretically the superconvergent fourth‐order accuracy in the first‐order partial derivatives of the collocation solution at the partition nodes. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 285–303, 1999     

A Legendre spectral quadrature Galerkin method for the Cauchy-Navier equations of elasticity with variable coefficients

We solve the Dirichlet and mixed Dirichlet-Neumann boundary value problems for the variable coefficient Cauchy-Navier equations of elasticity in a square using a Legendre spectral Galerkin method. The resulting linear system is solved by the preconditioned conjugate gradient (PCG) method with a preconditioner which is shown to be spectrally equivalent to the matrix of the resulting linear system. Numerical tests demonstrating the convergence properties of the scheme and PCG are presented.     

An alternating direction implicit method for orthogonal spline collocation linear systems

Summary An Alternating Direction Implicit method is analyzed for the solution of linear systems arising in high-order, tensor-product orthogonal spline collocation applied to some separable, second order, linear, elliptic partial differential equations in rectangles. On anNxN partition, with Jordan's selection of the acceleration parameters, the method requiresO(N ² ln ² N) arithmetic operations to produce an approximation whose accuracy, in theH ¹-norm, is that of the collocation solution.     

Self-assembled growth of nanostructural Ge islands on bromine-passivated Si(111) surfaces at room temperature

We have deposited relatively thick (∼60 nm) Ge layers on Br-passivated Si(111) substrates by thermal evaporation under high vacuum conditions at room temperature. Ge has grown in a layer-plus-island mode although it is different from the Stranski-Krastanov growth mode observed in epitaxial growth. Both the islands and the layer are nanocrystalline. This appears to be a consequence of reduction of surface free energy of the Si(111) substrate by Br-passivation. The size distribution of the Ge nanoislands has been determined. The Br-Si(111) substrates were prepared by a liquid treatment, which may not produce exactly reproducible surfaces. Nevertheless, some basic features of the nanostructural island growth are reasonably reproducible, while there are variations in the details of the island size distribution.     

Modified nodal cubic spline collocation for three-dimensional variable coefficient second order partial differential equations

We formulate a fourth order modified nodal cubic spline collocation scheme for variable coefficient second order partial differential equations in the unit cube subject to nonzero Dirichlet boundary conditions. The approximate solution satisfies a perturbed partial differential equation at the interior nodes of a uniform $N\times N\times N$ partition of the cube and the partial differential equation at the boundary nodes. In the special case of Poisson’s equation, the resulting linear system is solved by a matrix decomposition algorithm with fast Fourier transforms at a cost $O(N^3\log N)$ . For the general variable coefficient diffusion-dominated case, the system is solved using the preconditioned biconjugate gradient stabilized method.     

Hydroxycarbonyl anion (m/z 45), a diagnostic marker for α‐hydroxy carboxylic acids

Collision‐induced dissociation mass spectra of anions derived from α‐hydroxy carboxylic acids (AHAs) show a diagnostic peak at m/z 45. Product ion spectra recorded from this m/z 45 ion confirm that it represents the hydroxycarbonyl anion ( ), and not the formate anion ( ) as sometimes described in the literature. For example, the formate anion is not only defiant to further fragmentation but is also unreactive toward CO₂. In contrast, the hydroxycarbonyl anion easily fragments to produce a peak at m/z 17 for the hydroxyl anion, and also readily reacts with CO₂ to produce a peak at m/z 61 for the bicarbonate anion. The hydrogen atom in the hydroxycarbonyl anion and that in the formate anion are not mobile within the skeletal framework of the ions, since the two ions did not manifest any interconversion under the conditions and time scales of our mass spectrometric experiments. The other significant product ion peak in the spectra of deprotonated AHAs represents a 46‐Da loss. MS/MS data from appropriately deuteriated compounds confirmed that one hydrogen atom from the C‐2 position, and the other from the hydroxy group are specifically removed for this loss of elements of formic acid. Moreover, the two oxygen atoms eliminated for the HCOOH loss originate exclusively from the carboxylate group. Copyright © 2008 John Wiley & Sons, Ltd.     

Legendre Gauss Spectral Collocation for the Helmholtz Equation on a Rectangle

A spectral collocation method with collocation at the Legendre Gauss points is discussed for solving the Helmholtz equation –u+(x,y)u=f(x,y) on a rectangle with the solution u subject to inhomogeneous Robin boundary conditions. The convergence analysis of the method is given in the case of u satisfying Dirichlet boundary conditions. A matrix decomposition algorithm is developed for the solution of the collocation problem in the case the coefficient (x,y) is a constant. This algorithm is then used in conjunction with the preconditioned conjugate gradient method for the solution of the spectral collocation problem with the variable coefficient (x,y).     

An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolicinitial-boundary value problem involving a mixed derivativeterm, continuous variable coefficients and non-homogeneous Dirichletboundary conditions. In comparison to the alternating directionimplicit Laplace-modified method of Fernandes (1997), we formulateand analyse a new parameter-free alternating direction implicitscheme in which the standard central difference formula is usedfor the time approximation and orthogonal spline collocationis used for the spatial discretization. We establish unconditionalstability of the scheme, and its optimal order in the discretemaximum norm in time and the H¹ norm in space. Numerical experimentsindicate that the new scheme, which has the same order as themethod of Fernandes (1997, Numer. Math., 77, 223–241),is more accurate. We also show that the new scheme is easilygeneralized to the second-order hyperbolic problems on rectangularpolygons. Extensions of the scheme to problems with discontinuouscoefficients, nonlinear problems, and problems with other boundaryconditions are also discussed.