The finite element method for a boundary value problem with strong singularity

The existence and uniqueness of the R_ν-generalized solution for the third-boundary-value problem and the non-self-adjoint second-order elliptic equation with strong singularity are established. We construct a finite element method with a basis containing singular functions. The rate of convergence of the approximate solution to the R_ν-generalized solution in the norm of the Sobolev weighted space is established and, finally, results of numerical experiments are presented.     

Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations

In this paper, a new stabilized finite volume method is studied and developed for the stationary Navier-Stokes equations. This method is based on a local Gauss integration technique and uses the lowest equal order finite element pair P ₁–P ₁ (linear functions). Stability and convergence of the optimal order in the H ¹-norm for velocity and the L ²-norm for pressure are obtained. A new duality for the Navier-Stokes equations is introduced to establish the convergence of the optimal order in the L ²-norm for velocity. Moreover, superconvergence between the conforming mixed finite element solution and the finite volume solution using the same finite element pair is derived. Numerical results are shown to support the developed convergence theory.     

A posteriori error estimates for the coupling equations of scalar conservation laws

In this paper we prove a posteriori L ₂(L ₂) and L _∞(H ^?1) residual based error estimates for a finite element method for the one-dimensional time dependent coupling equations of two scalar conservation laws. The underlying discretization scheme is Characteristic Galerkin method which is the particular variant of the Streamline diffusion finite element method for δ=0. Our estimate contains certain strong stability factors related to the solution of an associated linearized dual problem combined with the Galerkin orthogonality of the finite element method. The stability factor measures the stability properties of the linearized dual problem. We compute the stability factors for some examples by solving the dual problem numerically.     

The products of conjugacy classes in some infinite simple groups

LetH _ν =S/S ^ν , whereS is the group of all permutations of a set of cardinality ? _ν andS ^v is its subgroup of permutations moving less than ? _ν elements. The infinite simple groupsH _ν ,ν>0, have covering number two; that is,C ²=H ^ν holds for each nonunit conjugacy classC[M]. Janko’s small groupJ ₁, the only finite simple group with covering number two, satisfies also: 1 $$C_{^1 } \subseteq C_{^2 } \cdot C_{^3 } for any nonunit classes C_{^1 } ,C_{^2 } ,C_{^3 } $$ . In fact,H _ν (ν>0) are the only groups of covering number two where (*) is known to fail. In this paper we determine arbitrary products of classes inH _ν (ν>0).     

A Note on the Complexity of Solving Poisson's Equation for Spaces of Bounded Mixed Derivatives

We study the complexity of solving the d-dimensional Poisson equation on ]0, 1[^d. We restrict ourselves to cases where the solution u lies in some space of functions of bounded mixed derivatives (with respect to the L_∞- or the L₂-norm) up to ∂^2d/∏^d_j=1 ∂x²_j. An upper bound for the complexity of computing a solution of some prescribed accuracy ε with respect to the energy norm is given, which is proportional to ε⁻¹. We show this result in a constructive manner by proposing a finite element method in a special sparse grid space, which is obtained by an a priori grid optimization process based on the energy norm. Thus, the result of this paper is twofold: First, from a theoretical point of view concerning the complexity of solving elliptic PDEs, a strong tractability result of the order O(ε⁻¹) is given, and, second, we provide a practically usable hierarchical basis finite element method of this complexity O(ε⁻¹), i.e., without logarithmic terms growing exponentially in d, at least for our sparse grid setting with its underlying smoothness requirements.     

A new mixed finite element method based on the Crank-Nicolson scheme for Burgers’ equation

In this paper, a new mixed finite element method is used to approximate the solution as well as the flux of the 2D Burgers’ equation. Based on this new formulation, we give the corresponding stable conforming finite element approximation for the P₀² ? P₁ pair by using the Crank-Nicolson time-discretization scheme. Optimal error estimates are obtained. Finally, numerical experiments show the efficiency of the new mixed method and justify the theoretical results.     

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(τ+h^l+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(τ²+h^l+1).     

Equazioni monodimensionali di un gas viscoso barotropico con una perturbazione poco regolare

The equations for a barotropic viscous gas in one space dimensiondν=(μ(?ν_ε)_ε?p _ε)dt+dG,? _t +?²ν_ε=0,p=?^γ with a perturbationdG are considered under the assumption thatG is only a function of bounded variation inL ²(Θ) orH ₀ ¹ (Θ) (Θ=]0, α[) and the esistence and the uniqueness of the global solution in a class of solutions of «strong type» as well as in a class of solutions of «weak type» are proved. This result constitutes a generalization of the result of Kazhikhov [8] and that of Shelukhin [10] and contains preliminary considerations for the corrisponding stochastic equations.     

The finite element method for a degenerate hyperbolic partial differential equation

Cahlon has recently considered finite difference methods for the degenerate Cauchy problem (tu _t)_t=Δu,u(x, 0)=u ₀(x) withu ₀ analytic. In this paper existence of a unique solution is shown for a more general class of initial data. A smoothing property of the solution operator is then exhibited. The usual semidiscrete finite element method is considered. The approximation is shown to be stable and superconvergent with orderO(h ^2μ?2) inl ₂ andl _∞, whereμ?1 is the degree of the polynomials used. OptimalL ² andL ^∞ estimates are also derived.     

Eigenfunction expansions for the Schrödinger equation with an inverse-square potential

We consider the one-dimensional Schrödinger equation -f″ + q_κf = Ef on the positive half-axis with the potential q_κ(r) = (κ² - 1/4)r^-2. For each complex number ν, we construct a solution u_ν^κ(E) of this equation that is analytic in κ in a complex neighborhood of the interval (-1, 1) and, in particular, at the “singular” point κ = 0. For -1 < κ < 1 and real ν, the solutions u_ν^κ(E) determine a unitary eigenfunction expansion operator U_κ,ν: L₂(0,∞) → L₂(R, Vκ,ν), where Vκ,ν is a positive measure on R. We show that every self-adjoint realization of the formal differential expression -?_r² + q_κ(r) for the Hamiltonian is diagonalized by the operator U_κ,ν for some ν ∈ R. Using suitable singular Titchmarsh–Weyl m-functions, we explicitly find the measures V_κ,ν and prove their continuity in κ and ν.     

Wavelet characterization of growth spaces of harmonic functions

We consider the space h _ν ^∞ of harmonic functions in R ₊ ⁿ⁺¹ with finite norm ‖u‖ _ν = sup |u(x, t)|/v(t), where the weight ν satisfies the doubling condition. Boundary values of functions in h _ν ^∞ are characterized in terms of their smooth multiresolution approximations. The characterization yields the isomorphism of Banach spaces h _ν ^∞ ～ l ^∞. The results are also applied to obtain the law of the iterated logarithm for the oscillation of functions in h _ν ^∞ along vertical lines.     

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L ² and the H ¹ norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.     

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L² and H¹ norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.     

Explicit solution of the inverse kinematic problem in the non-Herglotz case

The inverse kinematic problem is solved in the half space R ₊ ^ν+1 ={(x,z)|z?0,x∈R^ν, ν?1 under the assumption that the index of refraction can be represented in the form $$n^2 (x,z) = K^2 (z) + \sum\limits_{j = 1}^\nu {\Phi _j^2 (x_j ),} n_z< 0.$$ . The solution obtained is a generalization of the Herglotz-Wiechert formula. A formula is presented for the solution of the inverse kinematic problem in the general case of separation of variables in the eikonal equation.     

The disengagement algorithm or a new generalization of the exclusion algorithm

Let us consider a finite inf semilattice G with a set 1 of internal binary operations 1₁, isotono, satisfying certain conditions of no dispersion, of increasing and of substitution, and so that the greatest lower bound is distributive relatively to 1₁. A finite subset A of G being given, this article gives a method for enumerating the maximal elements of the sub-algebra A¹ generated by A with regard to 1, when A¹ is finite. This method, called disengagement algorithm, allows to examine each element once; it generalizes an algorithm giving the maximal n-rectangles of a part of a product of distributive lattices algorithm which already generalized a conjecture of Tison in Boolean algebra. Two applications are developed.     

C 0 elements for generalized indefinite Maxwell equations

In this paper we develop the C ⁰ finite element method for a generalized curlcurl-grad div indefinite Maxwell problem in a Lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the H ^r regularity for some r?<?1. The ingredients of our method are that two ??mass-lumping?? L ² projectors are applied to curl and div operators in the problem and that C ⁰ linear element or isoparametric bilinear element enriched with one element-bubble on every triangle element or with two-element-bubbles on every quadrilateral element, respectively, is employed for each component of the nonsmooth solution. Due to the fact that the element-bubbles can be statically eliminated at element levels, our method is essentially three-nodes or four-nodes C ⁰ Lagrange element method. With two Fortin-type interpolations established, a very technical duality argument is elaborated to estimate the error for the indefinite problem. For the nonsmooth solution having the H ^r regularity where r may vary in the interval [0, 1), we obtain the error bound ${{\mathcal O}(h^r)}$ in an energy norm. Some numerical experiments are performed to confirm the theoretical error bounds.     

Lattices generated by joins of the flats in orbits under finite affine-singular symplectic group and its characteristic polynomials

Let ASG(2ν + l, ν;F _q ) be the (2ν + l)-dimensional affine-singular symplectic space over the finite field F _q and ASp_2ν+l,ν (F _q ) be the affine-singular symplectic group of degree 2ν + l over F _q . Let O be any orbit of flats under ASp_2ν+l,ν (F _q ). Denote by L ^J the set of all flats which are joins of flats in O such that O ? L ^J and assume the join of the empty set of flats in ASG(2ν + l, ν;F _q ) is ?. Ordering L ^J by ordinary or reverse inclusion, then two lattices are obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice L ^J , when the lattices form geometric lattice, lastly gives the characteristic polynomial of L ^J .     

A discrete “three-particle” Schrödinger operator in the Hubbard model

In the space L ₂(T ^ν ×T ^ν ), where T ^ν is a ν-dimensional torus, we study the spectral properties of the “three-particle” discrete Schrödinger operator ? = H₀ + H₁ + H₂, where H₀ is the operator of multiplication by a function and H₁ and H₂ are partial integral operators. We prove several theorems concerning the essential spectrum of ?. We study the discrete and essential spectra of the Hamiltonians H^t and h arising in the Hubbard model on the three-dimensional lattice.     

A priori error analysis of the mortar finite element method for variational inequalities

The mortar finite element method is a special domain decomposition method, which can handle the situation where meshes on different subdomains need not align across the interface. In this article, we will apply the mortar element method to general variational inequalities of free boundary type, such as free seepage flow, which may show different behaviors in different regions. We prove that if the solution of the original variational inequality belongs to H²(D), then the mortar element solution can achieve the same order error estimate as the conforming P₁ finite element solution. Application of the mortar element method to a free surface seepage problem and an obstacle problem verifies not only its convergence property but also its great computational efficiency. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Two-level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. Based on the P₁-P₁ triangular element and using the pressure projection stabilized finite element method, we solve a small Navier-Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H²), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.