AC 0-collocation-like method for two-point boundary value problems

Summary A method of a collocation type based onC ⁰-piecewise polynomial spaces is presented for a two-point boundary value problem of the second order. The method has an optimal order of convergence under smoothness requirements on the exact solution which are weaker than forC ¹-collocation methods. If the differential operator is symmetric, a modification of this method leads to a symmetric system of linear equations. It is shown that if the collocation solution is a piecewise polynomial of degree not greater thanr, the method is stable and convergent with orderh ^r inH ¹-norm. A similar symmetric modification forC ⁰-colloction-finite element method [7] is also obtained. Superconvergence at the nodes is established.     

On the approximate calculation of time-minimal distributed controls of vibrations by finite element approximations

This paper is concerned with distributed null-control of vibrations governed by an abstract wave equation. Based on a method for the exact computation of minimumL ²-norm controls for given time intervals and time-minimal controls which are bounded with respect to theL ²-norm, an approximation method is developed which is based on Galerkin's method ana convergence results are derived.This paper is based on U. Lamp's doctoral dissertation and was supported by the Deutsche Forschungsgemeinschaft.     

Solving the equation −uxx − ϵuyy = f(x,y, u) by an O(h4) finite difference method

The semi‐linear equation −u_xx − ϵu_yy = f(x, y, u) with Dirichlet boundary conditions is solved by an O(h⁴) finite difference method, which has local truncation error O(h²) at the mesh points neighboring the boundary and O(h⁴) at most interior mesh points. It is proved that the finite difference method is O(h⁴) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 395–407, 2000     

Error analysis of the finite-strip method for parabolic equations

The finite-strip method (FSM) is a hybrid technique which combines spectral and finite-element methods. Finite-element approximations are made for each mode of a finite Fourier series expansion. The Galerkin formulated method is set apart from other weighted-residual techniques by the selection of two types of basis functions, a piecewise linear interpolating function and a trigonometric function. The efficiency of the FSM is due in part to the orthogonality of the complex exponential basis: The linear system which results from the weak formulation is decoupled into several smaller systems, each of which may be solved independently. An error analysis for the FSM applied to time-dependent, parabolic partial differential equations indicates the numerical solution error is O(h² + M^?r). M represents the Fourier truncation mode number and h represents the finite-element grid mesh. The exponent r ≥ 2 increases with the exact solution smoothness in the respective dimension. This error estimate is verified computationally. Extending the result to the finite-layer method, where a two-dimensional trigonometric basis is used, the numerical solution error is O(h² + M^?r + N^?q). The N and q represent the trucation mode number and degree of exact solution smoothness in the additional dimension. © 1993 John Wiley & Sons, Inc.     

An optimal order multigrid method for biharmonic,C 1 finite element equations

Summary In this paper, we study a special multigrid method for solving large linear systems which arise from discretizing biharmonic problems by the Hsieh-Clough-Tocher,C ¹ macro finite elements or several otherC ¹ finite elements. Since the multipleC ¹ finite element spaces considered are not nested, the nodal interpolation operator is used to transfer functions between consecutive levels in the multigrid method. This method converges with the optimal computational order.     

Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions

We present a new method, called UTA^GMS, for multiple criteria ranking of alternatives from set A using a set of additive value functions which result from an ordinal regression. The preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives A^R ⊆ A, called reference alternatives. The preference model built via ordinal regression is the set of all additive value functions compatible with the preference information. Using this model, one can define two relations in the set A: the necessary weak preference relation which holds for any two alternatives a, b from set A if and only if for all compatible value functions a is preferred to b, and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b. These relations establish a necessary and a possible ranking of alternatives from A, being, respectively, a partial preorder and a strongly complete relation. The UTA^GMS method is intended to be used interactively, with an increasing subset A^R and a progressive statement of pairwise comparisons. When no preference information is provided, the necessary weak preference relation is a weak dominance relation, and the possible weak preference relation is a complete relation. Every new pairwise comparison of reference alternatives, for which the dominance relation does not hold, is enriching the necessary relation and it is impoverishing the possible relation, so that they converge with the growth of the preference information. Distinguishing necessary and possible consequences of preference information on the complete set of actions, UTA^GMS answers questions of robustness analysis. Moreover, the method can support the decision maker when his/her preference statements cannot be represented in terms of an additive value function. The method is illustrated by an example solved using the UTA^GMS software. Some extensions of the method are also presented.     

New quasi-Newton method for solving systems of nonlinear equations

We propose a new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR or LU decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires O(n ²) arithmetic operations per iteration in contrast with the Newton method, which requires O(n ³) operations per iteration. Computational experiments confirm the high efficiency of the new method.     

Quintic trigonometric spline based numerical scheme for nonlinear modified Burgers' equation

This paper presents a numerical method based on quintic trigonometric B‐splines for solving modified Burgers' equation (MBE). Here, the MBE is first discretized in time by Crank–Nicolson scheme and the resulting scheme is solved by quintic trigonometric B‐splines. The proposed method tackles nonlinearity by using a linearization process known as quasilinearization. A rigorous analysis of the stability and convergence of the proposed method are carried out, which proves that the method is unconditionally stable and has order of convergence O(h⁴ + k²). Numerical results presented are very much in accordance with the exact solution, which is established by the negligible values of L₂ and L_∞ errors. Computational efficiency of the scheme is proved by small values of CPU time. The method furnishes results better than those obtained by using most of the existing methods for solving MBE.     

The Qualocation Method for Symm's Integral Equation on a Polygon

This paper discusses the convergence of the qualocation method for Symm's integral equation on closed polygonal boundaries in IR². Qualocation is a Petrov-Galerkin method in which the outer integrals are performed numerically by special quadrature rules. Before discretisation a nonlinear parametrisation of the polygon is introduced which varies more slowly than arc-length near each corner and leads to a transformed integral equation with a regular solution. We prove that the qualocation method using smoothest splines of any order k on a uniform mesh (with respect to the new parameter) converges with optimal order O (h^k). Furthermore, the method is shown to produce superconvergent approximations to linear functionals, retaining the same high convergence rates as in the case of a smooth curve.     

Analysis of a fourth-order compact ADI method for a linear hyperbolic equation with three spatial variables

This paper is concerned with a three-level alternating direction implicit (ADI) method for the numerical solution of a 3D hyperbolic equation. Stability criterion of this ADI method is given by using von Neumann method. Meanwhile, it is shown by a discrete energy method that it can achieve fourth-order accuracy in both time and space with respect to H ¹- and L ²-norms only if stable condition is satisfied. It only needs solution of a tri-diagonal system at each time step, which can be solved by multiple applications of one-dimensional tri-diagonal algorithm. Numerical experiments confirming the high accuracy and efficiency of the new algorithm are provided.     

A stabilizer-free C0 weak Galerkin method for the biharmonic equations

In this article, we present and analyze a stabilizer-free C⁰ weak Galerkin (SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is formulated in terms of cell unknowns which are C⁰ continuous piecewise polynomials of degree k + 2 with k ≽ 0 and in terms of face unknowns which are discontinuous piecewise polynomials of degree k + 1. The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the C¹ conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete H²-like norm and the H¹ norm for k ≽ 0 are established for the corresponding WG finite element solutions. Error estimates in the L² norm are also derived with an optimal order of convergence for k > 0 and sub-optimal order of convergence for k = 0. Numerical experiments are shown to confirm the theoretical results.

     

An L2 finite element approximation for the incompressible Navier–Stokes equations

This paper utilizes the Picard method and Newton's method to linearize the stationary incompressible Navier–Stokes equations and then uses an LL* approach, which is a least-squares finite element method applied to the dual problem of the corresponding linear system. The LL* approach provides an L²-approximation to a given problem, which is not typically available with conventional finite element methods for nonlinear second-order partial differential equations. We first show that the proposed combination of linearization scheme and LL* approach provides an L²-approximation to the stationary incompressible Navier–Stokes equations. The validity of L²-approximation is proven through the analysis of the weak problem corresponding to the linearized Navier–Stokes equations. Then, the convergence is analyzed, and numerical results are presented.     

Lower bounds for increasing complexity of derivations after cut elimination

We denote by C _k ^* the formula. In this paper for all k there is constructed a derivation of C _k ^* with cut, the number of sequents in which depends linearly on k. On the other hand, it is impossible to give an upper bound which is a Kalmar elementary function of k for the number of sequents in any derivation of the formula C _k ^* without cuts, or for the number of disjunctions in a refutation by the method of resolutions of systems of disjunctions corresponding to the negation of the formula C _k ^* . In particular, one can construct a derivation with cut of the formula C ₆ ^* , in which there is contained no more than 253 sequents, but in seeking a derivation of C ₆ ^* by the method of resolutions it is necessary to construct more than 10¹⁹²⁰⁰ disjunctions. With the help of Skolemization and taking out of quantifiers with respect to the formula C _k ^* there is constructed a formula v₀B _k ⁺ (v₀), which satisfies the following conditions: 1) one can construct a derivation with cuts of the formula v₀B _k ⁺ (v₀) in the constructive predicate calculus, the number of sequents in which depends linearly on k; 2) it is impossible to give an upper bound which is a Kalmar elementary function of k of the length of a term t such that the formula B _k ⁺ (t) is derivable.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 137–161, 1979.     

Stability of penalty finite-element methods for nonconforming problems

Penalty methods have been proposed as a viable method for enforcing interelement continuity constraints on nonconforming elements. Particularly for fourth-order problems in which C¹-continuity leads to elements of high degree or complex composite elements, the use of penalty methods to enforce the C¹-continuity constraint appears promising. In this study we demonstrate equivalence of the finite-element penalty method to a hybrid method and provide a stability analysis which implies that the penalty method is stable only if reduced integration of a certain order is used. Numerical experiments confirm that the penalty method fails if this condition is not met.     

Natural convection in porous media ‐ a meshless solution of the Brinkman extended Darcy model

A numerical study is performed on steady natural convection inside a differentially heated square cavity. The cavity is filled with porous media which exhibits the Brinkman extended Darcy behavior. The solution procedure for coupled mass, momentum, and energy equations is based on primitive variables and RBF collocation method with r⁷ function. Numerical examples include calculations at filtration with Rayleigh number 100, and Darcy numbers 10^–3 and 10^–5. The solution is compared with reference results of the fine‐grid finite volume method.     

Computational existence proofs for spherical <Emphasis Type="Italic">t</Emphasis>-designs

Spherical t-designs provide quadrature rules for the sphere which are exact for polynomials up to degree t. In this paper, we propose a computational algorithm based on interval arithmetic which, for given t, upon successful completion will have proved the existence of a t-design with (t + 1)² nodes on the unit sphere S² ì \mathbbR³{S^2 \subset \mathbb{R}^3} and will have computed narrow interval enclosures which are known to contain these nodes with mathematical certainty. Since there is no theoretical result which proves the existence of a t-design with (t + 1)² nodes for arbitrary t, our method contributes to the theory because it was tested successfully for t = 1, 2, . . . , 100. The t-design is usually not unique; our method aims at finding a well-conditioned one. The method relies on computing an interval enclosure for the zero of a highly nonlinear system of dimension (t + 1)². We therefore develop several special approaches which allow us to use interval arithmetic efficiently in this particular situation. The computations were all done using the MATLAB toolbox INTLAB.     

Superconvergence of finite volume element method for a nonlinear elliptic problem

In this article, we consider the finite volume element method for the second‐order nonlinear elliptic problem and obtain the H¹ and W^1,∞ superconvergence estimates between the solution of the finite volume element method and that of the finite element method, which reveal that the finite volume element method is in close relationship with the finite element method. With these superconvergence estimates, we establish the L^p and W^1,p (2 < p ≤ ∞) error estimates for the finite volume element method for the second‐order nonlinear elliptic problem. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Smoothing noisy data with spline functions

Summary A procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m–1 fitted ton distinct, not necessarily equally spaced or uniformly weighted, data points is presented. The procedure requires orderm ² n operations and therefore permits efficient orderm ² n calculation of statistics associated with a polynomial smoothing spline, including the generalized cross validation. The method is a significant improvement over an existing method which requires ordern ³ operations.     

A compact C0 discontinuous Galerkin method for Kirchhoff plates

A compact C⁰ discontinuous Galerkin (CCDG) method is developed for solving the Kirchhoff plate bending problems. Based on the CDG (LCDG) method for Kirchhoff plate bending problems, the CCDG method is obtained by canceling the term of global lifting operator and enhancing the term of local lifting operator. The resulted CCDG method possesses the compact stencil, that is only the degrees of freedom belonging to neighboring elements are connected. The advantages of CCDG method are: (1) CCDG method just requires C⁰ finite element spaces; (2) the stiffness matrix is sparser than CDG (LCDG) method; and (3) it does not contain any parameter which can not be quantified a priori compared to C⁰ interior penalty (IP) method. The optimal order error estimates in certain broken energy norm and H¹‐norm for the CCDG method are derived under minimal regularity assumptions on the exact solution with the help of some local lower bound estimates of a posteriori error analysis. Some numerical results are included to verify the theoretical convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1265–1287, 2015     

Fourier–Finite-element Approximation of Elliptic Interface Problems in Axisymmetric Domains

The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations −p̂Δ₃û = f̂ in three-dimensional axisymmetric domains Ωˆ. Here, pˆ is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H¹(Ωˆ), if the interface is smooth or if it meets the boundary of Ωˆ at some edge. In general, the solution û contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ωˆ. The rate of convergence of the combined approximation in H¹(Ωˆ) is proved to be 𝒪(h+N⁻¹) (h, N: the parameters of the finite-element- and Fourier-approximation, with h→0, N→∞). The theoretical results are confirmed by numerical experiments.