On mixed methods for fourth-order problems

A mixed finite element method for the problem v + σ²Δ²v = x with different types of boundary conditions is described. The method converges, and it is well suited for the analysis of various evolution problems. The computation of the discrete solution is made by applying a sequence of iterative methods for block matrices: correspondingly to each iteration either a couple of Poisson problems or a couple of problems for the identity operator are solved, according to the value of the parameter σ. Some numerical results for two model examples are presented.     

A numerical method for exact boundary controllability problems for the wave equation

The computational approximation of exact boundary controllability problems for the wave equation in two dimensions is studied. A numerical method is defined that is based on the direct solution of optimization problems that are introduced in order to determine unique solutions of the controllability problem. The uniqueness of the discrete finite-difference solutions obtained in this manner is demonstrated. The convergence properties of the method are illustrated through computational experiments. Efficient implementation strategies for the method are also discussed. It is shown that for smooth, minimum L²-norm Dirichlet controls, the method results in convergent approximations without the need to introduce regularization. Furthermore, for the generic case of nonsmooth Dirichlet controls, convergence with respect to L² norms is also numerically demonstrated. One of the strengths of the method is the flexibility it allows for treating other controls and other minimization criteria; such generalizations are discussed. In particular, the minimum H¹-norm Dirichlet controllability problem is approximated and solved, as are minimum regularized L²-norm Dirichlet controllability problems with small penalty constants. Finally, a discussion is provided about the differences between our method and existing methods; these differences may explain why our methods provide convergent approximations for problems for which existing methods produce divergent approximations unless they are regularized in some manner.     

Preconditioned conjugate residual methods for the solution of spectral equations

Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. (The symmetric part of the coefficient matrix A is defined by (A + A^T)/2.) Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.     

Domain Decomposition Spectral Method for Mixed Inhomogeneous Boundary Value Problems of High Order Differential Equations on Unbounded Domains

In this paper, we develop domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations defined on unbounded domains. We introduce an orthogonal family of new generalized Laguerre functions, with the weight function x ^?? , ?? being any real number. The corresponding quasi-orthogonal approximation and Gauss-Radau type interpolation are investigated, which play important roles in the related spectral and collocation methods. As examples of applications, we propose the domain decomposition spectral methods for two fourth order problems, and the spectral method with essential imposition of boundary conditions. The spectral accuracy is proved. Numerical results demonstrate the effectiveness of suggested algorithms.     

Robust stabilizing regions of fractional-order PDμ controllers of time-delay fractional-order systems

This study investigates the robust stabilizing regions with stability degrees of fractional-order PD^μ controllers for time-delay fractional-order systems. By the D-decomposition technology, we identify the stabilizing regions by three types of curves, i.e., the real root boundary (RRB) curves, complex root boundary (CRB) curves and infinite root boundary (IRB) lines. The existence conditions and computing methods of RRB curves, CRB curves and IRB lines are proposed to determine the boundaries of the potential stabilizing regions. The Test Lines and the principle of the identifying the stabilizing regions are presented to find the real stabilizing regions with a given stability degree. To deal with noises existing in the feedback signals, fractional-order PD^μ controllers involving filers are adopted. Meanwhile, the robust stabilizing regions are also analyzed via IRB curves, CRB curves and IRB lines with stability degrees. Finally, some illustrative examples are offered to verify the effectiveness of depicting algorithms of the robust stabilizing regions for PD^μ controllers with no filer or filers, respectively.     

Superconvergence of coupling techniques in combined methods for elliptic equations with singularities

Several coupling techniques, such as the nonconforming constraints, penalty, and hybrid integrals, of the Ritz-Galerkin and finite difference methods are presented for solving elliptic boundary value problems with singularities. Based on suitable norms involving discrete solutions at specific points, superconvergence rates on solution derivatives are exploited by using five combinations, e.g., the nonconforming combination, the penalty combination, Combinations I and II, and symmetric combination. For quasi-uniform rectangular grids, the superconvergence rates, O(h^2−δ), of solution derivatives by all five combinations can be achieved, where h is the maximal mesh length of difference grids used in the finite difference method, and δ(> 0) is an arbitrarily small number.Superconvergence analysis in this paper lies in estimates on error bounds caused by the coupling techniques and their incorporation with finite difference methods. Therefore, a similar analysis and conclusions may be extended to linear finite element methods using triangulation by referring to existing references. Moreover, the five combinations having O(h^2−δ) of solution derivatives are well suited to solving engineering problems with multiple singularities and multiple interfaces.     

A nonconforming finite element method for a singularly perturbed boundary value problem

We analyze a new nonconforming Petrov-Galerkin finite element method for solving linear singularly perturbed two-point boundary value problems without turning points. The method is shown to be convergent, uniformly in the perturbation parameter, of orderh ^1/2 in a norm slightly stronger than the energy norm. Our proof uses a new abstract convergence theorem for Petrov-Galerkin finite element methods.     

Eigenfrequency determination for arbitrary cross-section waveguides

In this paper, we outline the construction of Maple routines for the solution of the Helmholtz equation ▿²ψ + k²ψ = 0 with Dirichlet boundary conditions in two-dimensional domains. By means of the symbolic manipulator, we are able to perform a numerical study of the eigenvalues for quantum billiards.     

Adaptive quadratures over volumes

We consider the numerical approximation of volume integrals over bounded domainsD:={D∈R ³:H(x>≤0}, whereH:R ³→R is a suitable decidability function. The integrands may be smooth maps or singular maps such as those arising in the volume potentials for boundary integral methods. An adaptive extrapolation method is described which is based on some simple quadrature rules. It utilizes an automatic simplicial subdivision of the domain. The method offers improvements over recently given approaches. A special version is offered for the important application of the numerical computation of volume potentials in boundary integral methods. Several examples illustrate the performance of the method.     

Coupling strategy for matching different methods in solving singularity problems

For solving elliptic boundary value problems with singularities, we have proposed the combined methods consisting of the Ritz-Galerkin method using singular (or analytic) basic functions for one part,S ₂, of the solution domainS, where there exist singular points, and the finite element method for the remaining partS ₁ ofS, where the solution is smooth enough. In this paper, general approaches using additional integrals are presented to match different numerical methods along their common boundary Г₀. Errors and stability analyses are provided for such a general coupling strategy. These analyses are important because they form a theoretical basis for a number of combinations between the Ritz-Galerkin and finite element methods addressed in [7], and because they can lead to new combinations of other methods, such as the combined methods of the Ritz-Galerkin and finite difference methods. Moreover, the analyses in this paper can be applied or extended to solve general elliptic boundary value problems with angular singularities, interface singularity or unbounded domain.     

A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems

We analyze the discontinuous finite element errors associated with p-degree solutions for two-dimensional first-order hyperbolic problems. We show that the error on each element can be split into a dominant and less dominant component and that the leading part is O(h^p+1) and is spanned by two (p+1)-degree Radau polynomials in the x and y directions, respectively. We show that the p-degree discontinuous finite element solution is superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree Radau polynomial. For a linear model problem, the p-degree discontinuous Galerkin solution flux exhibits a strong O(h^2p+2) local superconvergence on average at the element outflow boundary. We further establish an O(h^2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct simple, efficient and asymptotically correct a posteriori finite element error estimates for multi-dimensional first-order hyperbolic problems in regions where solutions are smooth.     

An efficient solution algorithm for boundary element equations

Boundary element techniques result in the solution of a linear system of equations of the type HU = GQ + B, which can be transformed into a system of equations of the type AX = F. The coefficient matrix A requires the storage of a full matrix on the computer. This storage requirement, of the order of n^*n memory positions (n = number of equations), for a very large n is often considered negative for the boundary element method. Here, two algorithms are presented where the memory requirements to solve the system are only n^*(n - 1)/2 and n^*n/4 respectively. The algorithms do not necessitate any external storage devices nor do they increase the computational efforts.     

Improvement of the accuracy in boundary element method based on high-order discretization

The boundary element method (BEM) is a popular method to solve various problems in engineering and physics and has been used widely in the last two decades. In high-order discretization the boundary elements are interpolated with some polynomial functions. These polynomials are employed to provide higher degrees of continuity for the geometry of boundary elements, and also they are used as interpolation functions for the variables located on the boundary elements. The main aim of this paper is to improve the accuracy of the high-order discretization in the two-dimensional BEM. In the high-order discretization, both the geometry and the variables of the boundary elements are interpolated with the polynomial function P_m, where m denotes the degree of the polynomial. In the current paper we will prove that if the geometry of the boundary elements is interpolated with the polynomial function P_m+1 instead of P_m, the accuracy of the results increases significantly. The analytical results presented in this work show that employing the new approach, the order of convergence increases from O(L₀)^m to O(L₀)^m+1 without using more CPU time where L₀ is the length of the longest boundary element. The theoretical results are also confirmed by some numerical experiments.     

IA*: An adjacency-based representation for non-manifold simplicial shapes in arbitrary dimensions

We propose a compact, dimension-independent data structure for manifold, non-manifold and non-regular simplicial complexes, that we call the Generalized Indexed Data Structure with Adjacencies (IA^?data structure). It encodes only top simplices, i.e. the ones that are not on the boundary of any other simplex, plus a suitable subset of the adjacency relations. We describe the IA^? data structure in arbitrary dimensions, and compare the storage requirements of its 2D and 3D instances with both dimension-specific and dimension-independent representations. We show that the IA^? data structure is more cost effective than other dimension-independent representations and is even slightly more compact than the existing dimension-specific ones. We present efficient algorithms for navigating a simplicial complex described as an IA^? data structure. This shows that the IA^? data structure allows retrieving all topological relations of a given simplex by considering only its local neighborhood and thus it is a more efficient alternative to incidence-based representations when information does not need to be encoded for boundary simplices.     

Minimal quadrature formulae for the spaces H 2 R and L 2 R . A unified interpolatory approach

We consider minimal quadrature formulae for the Hilbert spacesH ₂ ^R andL ₂ ^R consisting of functions which are analytical on the open disc with radiusR and centre at the origin; the inner products are the boundary contour integral forH ₂ ^R and the area integral over the disc forL ₂ ^R . Such formulae can be viewed as interpolatory, generalizing—in two ways—Markoff's idea to construct the classical Gaussian quadratur formulae. This can be done simultaneously for both spaces using the same Hermitian interpolating operator. The advantage of this approach to minimal formulae is that we get a nonlinear system of equations for the nodes of the minimal formulae alone, in contrast to the coupled system for nodes and weights which arises from the minimality conditions. The uncoupled system that we obtain is numerically solvable for reasonable numbers of nodes and numerical tests show that the resulting minimal formulae are very well suited for the integration of functions with boundary singularities.     

A strictly conservative Cartesian cut-cell method for compressible viscous flows on adaptive grids

A Cartesian cut-cell method which allows the solution of two- and three-dimensional viscous, compressible flow problems on arbitrarily refined graded meshes is presented. The finite-volume method uses cut cells at the boundaries rendering the method strictly conservative in terms of mass, momentum, and energy. For three-dimensional compressible flows, such a method has not been presented in the literature, yet. Since ghost cells can be arbitrarily positioned in space the proposed method is flexible in terms of shape and size of embedded boundaries. A key issue for Cartesian grid methods is the discretization at mesh interfaces and boundaries and the specification of boundary conditions. A linear least-squares method is used to reconstruct the cell center gradients in irregular regions of the mesh, which are used to formulate the surface flux. Expressions to impose boundary conditions and to compute the viscous terms on the boundary are derived. The overall discretization is shown to be second-order accurate in L¹. The accuracy of the method and the quality of the solutions are demonstrated in several two- and three-dimensional test cases of steady and unsteady flows.     

Pressure projection stabilized finite element method for Navier–Stokes equations with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Navier–Stokes equation with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. The H ¹ and L ² error estimates for the velocity and the L ² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

Regularity properties of one-leg methods for delay differential equations

The main purpose of this paper is to investigate the asymptotic states of one-leg methods for multidelay differential equations. In particular, the existence of spurious steady solutions and period-2 solutions in constant or variable timestep is studied, and the concepts of R^[1]-regularity and R^[2]-regularity of one-leg methods for multidelay differential equations are introduced and studied. Some conditions guaranteeing R^[1]-regularity and R^[2]-regularity of such methods applied to multidelay differential equations with some important structures are given.     

Stability of Galerkin multistep procedures in time-homogeneous hyperbolic problems

Linear and time-homogeneous hyperbolic initial boundary value problems are approximated using Galerkin procedures for the space directions and linear multistep methods for the time direction. At first error bounds are proved for multistep methods having a stability interval [?ω, 0], 0<ω, and systemsY″=AY+C(t) under the condition that \(\Delta t^2 \left\| A \right\| \leqslant \omega \) Δt time step. Then these error bounds are applied to derive bounds for the error in hyperbolic problems. The result shows that the initial error and the discretization error grow liket andt ² respectively. But the initial error is multiplied with a factor which becomes large if the mesh width of the space discretization is small.