Compact High-Order Finite Differences for Interface Problems in One Dimension

This paper is concerned with the construction and analysis ofcompact finite difference approximations to the model linearsource problem –(pu')' + qu = f where the functions p,q, and f can have jump discontinuities at a finite number ofpoints. Explicit formulae that give O(h²) O(h³) and O(h⁴) accuracyare derived, and a procedure for computing three-point schemesof any prescribed order of accuracy is presented. A rigoroustruncation and discretization error analysis is offered. Numericalresults are also given.     

The streamline-diffusion method for nonconforming Qrot1 elements on rectangular tensor-product meshes

When the streamline–diffusion finite element method isapplied to convection–diffusion problems using nonconformingtrial spaces, it has previously been observed that stabilityand convergence problems may occur. It has consequently beenproposed that certain jump terms should be added to the bilinearform to obtain the same stability and convergence behaviouras in the conforming case. The analysis in this paper showsthat for the Q^rot₁ 1 element on rectangular shape-regular tensor-productmeshes, no jump terms are needed to stabilize the method. Inthis case moreover, for smooth solutions we derive in the streamline–diffusionnorm convergence of order h^3/2 (uniformly in the diffusion coefficientof the problem), where h is the mesh diameter. (This estimateis already known for the conforming case.) Our analysis alsoshows that similar stability and convergence results fail tohold true for analogous piecewise linear nonconforming elements.     

Periodicity in Group Cohomology and Complete Resolutions

A group G is said to have periodic cohomology with period qafter k steps, if the functors Hⁱ(G, –) and H^i+q(G, –)are naturally equivalent for all i > k. Mislin and the authorhave conjectured that periodicity in cohomology after some stepsis the algebraic characterization of those groups G that admita finite-dimensional, free G-CW-complex, homotopy equivalentto a sphere. This conjecture was proved by Adem and Smith underthe extra hypothesis that the periodicity isomorphisms are givenby the cup product with an element in H^q(G,Z). It is expectedthat the periodicity isomorphisms will always be given by thecup product with an element in H^q(G,Z); this paper shows thatthis is the case if and only if the group G admits a completeresolution and its complete cohomology is calculated via completeresolutions. It is also shown that having the periodicity isomorphismsgiven by the cup product with an element in H^q(G,Z) is equivalentto silp G being finite, where silp G is the supremum of theinjective lengths of the projective ZG-modules. 2000 MathematicsSubject Classification 20J05, 57S25.     

Supraconvergence of a finite difference scheme for solutions in Hs(0, L)

^** Email: silvia{at}mat.uc.pt^*** Email: ferreira{at}mat.uc.pt^**** Email: grigo{at}math.tu-berlin.de In this paper we study the convergence of a centred finite differencescheme on a non-uniform mesh for a 1D elliptic problem subjectto general boundary conditions. On a non-uniform mesh, the schemeis, in general, only first-order consistent. Nevertheless, weprove for s (1/2, 2] order O(h^s)-convergence of solution andgradient if the exact solution is in the Sobolev space H^1+s(0,L), i.e. the so-called supraconvergence of the method. It isshown that the scheme is equivalent to a fully discrete linearfinite-element method and the obtained convergence order isthen a superconvergence result for the gradient. Numerical examplesillustrate the performance of the method and support the convergenceresult.     

A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems

^** Email: emmanuil.georgoulis{at}mcs.le.ac.uk^*** Email: al{at}maths.strath.ac.uk We consider a variant of the hp-version interior penalty discontinuousGalerkin finite element method (IP-DGFEM) for second-order problemsof degenerate type. We do not assume uniform ellipticity ofthe diffusion tensor. Moreover, diffusion tensors of arbitraryform are covered in the theory presented. A new, refined recipefor the choice of the discontinuity-penalization parameter (thatis present in the formulation of the IP-DGFEM) is given. Makinguse of the recently introduced augmented Sobolev space framework,we prove an hp-optimal error bound in the energy norm and anh-optimal and slightly p-suboptimal (by only half an order ofp) bound in the L² norm (the latter, for the symmetric versionof the IP-DGFEM), provided that the solution belongs to an augmentedSobolev space.     

Increased Accuracy Cubic Spline Solutions to Two-Point Boundary Value Problems

The relation between finite difference approximation and cubicspline solutions of a two-point boundary value problem for thedifferential equation y' +f(x)y^'+g(x)y = r(x) has been consideredin a previous paper. The present paper extends the analysisto the integral equation formulation of the problem. It is shownthat an improvement in accuracy (local truncation error O(h⁶)rather than O(h⁴)) now results from a cubic spline approximationand that for the particular case f(x) 0 the resulting recurrencerelations have a form and accuracy similar to the well-knownNumerov formula. For this case also a formula with local truncationerror O(h⁸) is derived.     

A new approach to energy-based sparse finite-element spaces

^{We show that the logarithmic factor in the standard error estimatefor sparse finite element (FE) spaces in arbitrary dimensiond is removable in the energy (H1) norm. Via a penalized sparsegrid condition, we then propose and analyse a new version ofthe energy-based sparse FE spaces introduced first in Bungartz(1992, Dünne Gitter und deren Anwendung bei der adaptivenLösung der dreidimensionalen Poisson-Gleichung. Dissertation.Munich, Germany: TU München) and known to satisfy an optimalapproximation property in the energy norm.}     

Actions of Commutative Hopf Algebras

We show that actions of finite-dimensional semisimple commutativeHopf algebras H on H-module algebras A are essentially group-gradings.Moreover we show that the centralizer of H in the smash productA # H equals A^H H. Using these we invoke results about groupgraded algebras and results about centralizers of separablesubalgebras to give connections between the ideal structureof A, A^H and A # H. Examples of the above occur naturally when one considers: (1) finite abelian groups G of automorphisms of an algebra Awith | G |^–1 A; (2) G-graded algebras, for finite groups G; (3) finite-dimensional restricted Lie algebras L, with semisimplerestricted enveloping algebra u(L), acting as derivations onan algebra A.     

A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for thegeneral second-order non-linear differential equation y^N = f(x,y, y') subject to mixed two-point boundary conditions. An interestingfeature of our method is that each discretization of the differentialequation at an interior grid point is based on just three evaluationsof f. We establish, under appropriate conditions, O(h⁴)-convergenceof the finite difference scheme. In the case of linear differentialequations, our finite difference scheme leads to tridiagonallinear systems. Numerical examples are considered to demonstratecomputationally the fourth order of the method.     

Convergence rates for the coupling of FEM and collocation BEM

We prove convergence of the coupling of finite and boundaryelements where Galerkin's methd is used for finite elementsand collocation for boundary elements. We consider linear ellipticboundary value problems in two dimensions, in particular problemsin elasticity. The mesh width k of the boundary elements andthe mesh width h of the finite elements are required to satisfykßh with suitable ß. Asymptotic error estimatesin the energy norm and in the L²-norm are derived. Numericalexamples are included.     

A Sixth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems

We present a sixth-order finite difference method for the generalsecond-order non-linear differential equation Y"=f(x, y, y')subject to the boundary conditions y(a) = A, y(b) = B. In thecase of linear differential equations, our finite differencescheme leads to tridiagonal linear systems. We establish, underappropriate conditions, O(h⁶)-convergence of the finite differencescheme. Numerical examples are given to illustrate the methodand its sixth-order convergence.     

The Modulus of Smoothness and Discrete Data in a Square Domain

This paper relates to a function f on a two-dimensional squaredomain that is finite, infinite, or semi-infinite. It is shownthat, if the second difference of f with respect to a uniformsquare lattice of mesh-size 2^–n is uniformly of order2^–n with 0<<2, then the h-stepsize second differenceof f is uniformly of order h in any direction in the domain.This is deduced from a corresponding, more general weak-type(i.e. Marchaud-type) inequality.     

A Fourth-order Tridiagonal Finite Difference Method for General Two-point Boundary Value Problems with Non-linear Boundary Conditions

We present a fourth-order finite difference method for the generalsecond-order nonlinear differential equation y^" = f(x, y, y‘)subject to non-linear two-point boundary conditions g₁(y(a), — y^’()) = 0, g₂(y(b), y'(b)) = 0. When both the differential equation and the boundary conditionsare linear, the method leads to a tridiagonal linear system.We show that the finite difference method is O(h⁴)-convergent.Numerical examples are given to illustrate the method and itsfourth-order convergence. The present paper extends the methodgiven in Chawla (1978) to the case of non-linear boundary conditions.     

A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges

The solution of the Stokes problem in three-dimensional domainswith edges has anisotropic singular behaviour which is treatednumerically by using anisotropic finite element meshes. Thevelocity is approximated by Crouzeix–Raviart (nonconformingP₁ ) elements and the pressure by piecewise constants. Thismethod is stable for general meshes (without minimal or maximalangle condition). Denoting by N_e the number of elements in themesh, the interpolation and consistency errors are of the optimalorder h N_e^–1/3 which is proved for tensor product meshes.As a by-product, we analyse also nonconforming prismatic elementswith P₁ [oplus ] span {x₃²} as the local space for the velocitywhere x₃ is the direction of the edge.     

Semidiscrete finite element Galerkin approximations to the equations of motion arising in the Oldroyd model

In this paper, a semidiscrete finite element Galerkin methodfor the equations of motion arising in the 2D Oldroyd modelof viscoelastic fluids with zero forcing function is analysed.Some new a priori bounds for the exact solutions are derivedunder realistically assumed conditions on the data. Moreover,the long-time behaviour of the solution is established. By introducinga Stokes–Volterra projection, optimal error bounds forthe velocity in the L(L²) as well as in the L(H¹)-norms andfor the pressure in the L(L²)-norm are derived which are validuniformly in time t > 0.     

A qualocation method for a unidimensional single phase semilinear Stefan problem

Based on straightening the free boundary, a qualocation methodis proposed and analysed for a single phase unidimensional Stefanproblem. This method may be considered as a discrete versionof the H¹-Galerkin method in which the discretization is achievedby approximating the integrals by a composite Gauss quadraturerule. Optimal error estimates are derived in L(W^j,), j = 0,1,and L (H^j), j = 0,1,2, norms for a semidiscrete scheme withoutany quasi-uniformity assumption on the finite element mesh.     

Convergent and spurious solutions of nonlinear elliptic equations

In this paper we investigate finite element approximations ofnonlinear elliptic equations in three dimensions. By applyingand extending the results of Lopez-Marcos and Sanz-Serna, weprove that the finite element approximation on a mesh of sizeh, has a solution U_k which converges to an exact solution ofthe differential equation as h0. This solution is unique withina suitably defined stability ball Bh. For the particular nonlinearequation u + (u + u^p) we show that the size of B_h depends uponh only if p > 5 when it tends to zero as h 0. In this casewe prove the existence of spurious solutions V_h of the Galerkinapproximation which become unbounded in the maximum norm ash0. The stability ball B_h then acts to separate the convergentand the spurious solutions. We present the results of some numericalexperiments to substantiate our claims.     

A finite element method for a unidimensional single-phase nonlinear free boundary problem in groundwater flow

Consider a unidimensional, single-phase nonlinear Stefan problemwith nonlinear source and permeance terms, and a Dirichlet boundarycondition depending on the free boundary function. This problemis important in groundwater flow. By immobilizing the free boundarywith the help of a Landau-type transformation, together witha homogeneous transformation dealing with the nonhomogeneousDirichlet boundary condition, an H¹-finite element method forthe problem is proposed and analyzed. Global existence of theapproximate solution is established, and optimal error estimatesin L², L, H¹ and H² norms are derived for both semi-discreteand fully discrete schemes.     

A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

A new a posteriori L₂ norm error estimator is proposed for thePoisson equation. The error estimator can be applied to anisotropictetrahedral or triangular finite element meshes. The estimatoris rigorously analysed for Dirichlet and Neumann boundary conditions. The lower error bound relies on specifically designed anisotropicbubble functions and the corresponding inverse inequalities.The upper error bound utilizes non-standard anisotropic interpolationestimates. Its proof requires H² regularity of the Poisson problem,and its quality depends on how good the anisotropic mesh resolvesthe anisotropy of the problem. This is measured by a so-called‘matching function’. A numerical example supports the anisotropic error analysis.     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.