Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces

This paper considers the finite-element approximation of theelliptic interface problem: -?(u) + cu = f in Rⁿ (n = 2 or3), with u = 0 on , where is discontinuous across a smoothsurface in the interior of . First we show that, if the meshis isoparametrically fitted to using simplicial elements ofdegree k - 1, with k 2, then the standard Galerkin method achievesthe optimal rate of convergence in the H¹ and L² norms overthe approximations _l⁴ of _l where _{l 2}. Second, since itmay be computationally inconvenient to fit the mesh to , weanalyse a fully practical piecewise linear approximation ofa related penalized problem, as introduced by Babuska (1970),based on a mesh that is independent of . We show that, by choosingthe penalty parameter appropriately, this approximation convergesto u at the optimal rate in the H¹ norm over _l⁴ and in the L²norm over any interior domain _l^* satisfying _l^* _l^** _l⁴ for somedomain _l^**. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton BN1 9QH     

On spurious asymptotic numerical solutions of explicit Runge-Kutta methods

The bifurcation diagram associated with the logistic equationⁿ⁺¹ = aⁿ(1 – ⁿ) is by now well known, as is its equivalenceto solving the ordinary differential equation (ODE) u' = u(1– u) by the explicit Euler difference scheme. It has alsobeen noted by Iserles that other popular difference schemesmay not only exhibit period doubling and chaotic phenomena butalso possess spurious fixed points. We investigate, both analyticallyand computationally, Runge-Kutta schemes applied to the equationu'=f(u), for f(u) = u{1 – u) and f(u) = au(1 – u)(b– u), contrasting their behaviour with the explicit Eulerscheme. We determine and provide a local analysis of bifurcationsto spurious fixed points and periodic orbits. In particularwe show that these may appear below the linearised stabilitylimit of the scheme, and may consequently lead to erroneouscomputational results. Major part of the material was published as an internal report-NASATechnical Memorandum 102919, April 1990, also as Universityof Reading Numerical Analysis Report 3/90, March 1990. This work was performed whilst a visiting scientist at NASAAmes Research Center, Moffett Field. CA 94035 USA. Staff Scientist, Fluid Dynamics Division.     

Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient

We analyse approximate solutions generated by an upwind differencescheme (of Engquist–Osher type) for nonlinear degenerateparabolic convection–diffusion equations where the nonlinearconvective flux function has a discontinuous coefficient (x)and the diffusion function A(u) is allowed to be strongly degenerate(the pure hyperbolic case is included in our setup). The mainproblem is obtaining a uniform bound on the total variationof the difference approximation u, which is a manifestationof resonance. To circumvent this analytical problem, we constructa singular mapping (, ·) such that the total variationof the transformed variable z = (, u) can be bounded uniformlyin . This establishes strong L¹ compactness of z and, since(, ·) is invertible, also u. Our singular mapping isnovel in that it incorporates a contribution from the diffusionfunction A(u). We then show that the limit of a converging sequenceof difference approximations is a weak solution as well as satisfyinga Krukov-type entropy inequality. We prove that the diffusionfunction A(u) is Hölder continuous, implying that the constructedweak solution u is continuous in those regions where the diffusionis nondegenerate. Finally, some numerical experiments are presentedand discussed.     

Total Flux Estimates for a Finite-Element Approximation of Elliptic Equations

An elliptic boundary-value problem on a domain with prescribedDirichlet data on _I is approximated using a finite-elementspace of approximation power h^K in the L² norm. It is shownthat the total flux across _I can be approximated with an errorof O(h^K) when is a curved domain in Rⁿ (n = 2 or 3) and isoparametricelements are used. When is a polyhedron, an O(h^2K–2)approximation is given. We use these results to study the finite-elementapproximation of elliptic equations when the prescribed boundarydata on _I is the total flux. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton, Sussex BN1 9QH.     

Boundary integral methods for singularly perturbed boundary value problems

In this paper we consider boundary integral methods appliedto boundary value problems for the positive definite Helmholtz-typeproblem –U + ²U = 0 in a bounded or unbounded domain,with the parameter real and possibly large. Applications arisein the implementation of space–time boundary integralmethods for the heat equation, where is proportional to 1/(t),and t is the time step. The corresponding layer potentials arisingfrom this problem depend nonlinearly on the parameter and havekernels which become highly peaked as , causing standard discretizationschemes to fail. We propose a new collocation method with arobust convergence rate as . Numerical experiments on a modelproblem verify the theoretical results.     

Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces

We consider a fully practical finite-element approximationof the following system of nonlinear degenerate parabolic equations: (u)/(t) + . (u² [(v)]) - (1)/(3) .(u³ w)= 0, w = - c u - u^-+ a u^-3 , (v)/(t) + . (u v [(v)]) - v - .(u² v w) = 0. The above models a surfactant-driven thin-film flow in the presenceof both attractive, a>0, and repulsive, >0 with >3,van der Waals forces; where u is the height of the film, v isthe concentration of the insoluble surfactant monolayer and(v):=1-v is the typical surface tension. Here 0 and c>0 arethe inverses of the surface Peclet number and the modified capillarynumber. In addition to showing stability bounds for our approximation,we prove convergence, and hence existence of a solution to thisnonlinear degenerate parabolic system, (i) in one space dimensionwhen >0; and, moreover, (ii) in two space dimensions if inaddition 7. Furthermore, iterative schemes for solving the resultingnonlinear discrete system are discussed. Finally, some numericalexperiments are presented.     

The P-version Penalty Finite Element Method

A p-version penalty finite element method is used to solve themodel problem –u=f in , u=g on . Error estimates are derivedin H¹-norm. The p-version penalty method with extrapolationyields an approximate solution which converges at the optimalrate. Numerical results show the effectiveness of the p-versionpenalty method with extrapolation.     

A new non-conforming Petrov-Galerkin finite-element method with triangular elements for a singularly perturbed advection-diffusion problem

A new non-conforming exponentially fitted Petrov-Galerkin finite-elementmethod based on Delaunay triangulation and its Dirichlet tessellationis constructed for the numerical solution of singularly perturbedstationary advectiondiffusion problems with a singular perturbationparameter . The method is analyzed mathematically and its stabilityis shown to be independent of . The error estimate in an -independentdiscrete energy norm for the approximate solution is shown todepend on first-order seminorms of the flux and the zero-orderterm of the equation, the sup norm of the exact solution, thefirst-order seminorm of the coefficient of the advection term,and the approximation error of the inhomogeneous term. Sincethe first two seminorms are not bounded uniformly in , the -uniformconvergence of the method still remains an open question. Noassumption is required that the angles in the triangulationare all acute. Since the system matrix for this method is identicalto that for the exponentially fitted box method, the theoreticalresults also provide an analysis of that box method. The newmethod also contains the central-difference and upwind methodsas two limiting cases. It can be regarded as a weighted finite-differencemethod on a triangular mesh. Numerical results are presentedto show the superior performance of the method in comparisonwith the upwind and central-difference methods for a small increasein the computation cost. Present address: School of Mathematics, The University of NewSouth Wales, Kensington, NSW 2033, Australia.     

Discrete methods in the study of an inverse problem for Laplace's equation

Let u be harmonic in the interior of a rectangle and satisfythe third-kind boundary condition u_n + yu = where 0, y 0with supports included in the bottom and in the top side of, respectively. Recovering y from a knowledge of and of thetrace of u on the bottom is a nonlinear inverse problem ofinterest in the field of nondestructive evaluation. A convergentGalerkin method for approximating y is proposed and tested innumerical experiments.     

Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems

The long-time behaviour of continuous time Galerkin (CTG) approximationsof some well-known one-dimensional non-linear evolution problemswhich model phase transitions are analyzed. These numericalschemes are fully discrete and of arbitrary order. Partially supported by the Army Research Office through grant28535-MA. Partially supported by the ONR Contract No N00014-90-J-1238.     

Realistic a Priori and a Posteriori Error Bounds for Computed Eigenvalues

Permanent address: Department of Engineering Mathematics, Cairo University, Giza, Egypt. A priori and a posteriori error bounds are given for the computedeigenpair (, ) of the eigenvalue problem Ax = x, which are shownto be more realistic than some of the available ones. A simplemethod is also presented for computing the backward error. Finallya scaling procedure is explained for reducing the residual error.     

Mixed finite element analysis of a non-linear three-fields Stokes model

In this article, a mixed finite element analysis of the non-linearStokes problem with monotone constitutive laws is considered.We construct a new three-field model for incompressible fluidswhere the velocity u, the non-linear stress tensor = (|u|)u and the pressure p are the most relevant unknowns. We giveexistence and unicity results for the continuous problem andits approximation. Stable and optimal error estimates underminimal regularity assumptions are derived and numerical resultsare presented. Received 29 April 1999. Accepted 30 November 1999.     

A 'taut string algorithm' for straightening a piecewise linear path in two dimensions

A piecewise linear path in two dimensions is formed by drawingstraight lines between adjacent points in the sequence {u_i:i=1,2,...,n}.Let be a given positive number such that the length of eachstraight line segment is at least 3 . We straighten the pathin the following way. For i=2,3,...,n-1, we surround u_i by acircular ring of radius and centre u_i. Then a piece of stringthat begins at u₁ and ends at u_n is threaded through all therings in sequence. The new path is constructed by pulling thestring tight. An iterative algorithm is proposed that generatesthe new path to prescribed accuracy. Its convergence is provedand its efficiency is demonstrated by some numerical results.     

Finite-Element Approximation of a Plasma Equilibrium Problem

The plasma problem studied is: given R⁺ find (, d, u) R ?R ? H¹() such that Let ₁ < ₂ be the first two eigenvalues of the associatedlinear eigenvalue problem: find $$\left(\lambda ,\phi \right)\in\mathrm{R;}\times {\hbox{ H }}_{0}^{1}\left(\Omega \right)$$such that For ₀(0,₂) it is well known that there exists a unique solution(₀, d₀, u₀) to the above problem. We show that the standard continuous piecewise linear Galerkinfinite-element approximatinon $$\left({\lambda }_{0},{\hbox{d }}_{0}^{k},{u}_{0}^{h}\right)$$, for ₀(0,₂), converges atthe optimal rate in the H¹, L², and L norms as h, the mesh length,tends to 0. In addition, we show that dist (, ^h)Ch² ln 1/h,where $${\Gamma }^{\left(h\right)}=\left\{x\in \Omega :{u}_{0}^{\left(h\right)}\left(x\right)=0\right\}$$.Finally we consider a more practical approximation involvingnumerical integration.     

Set convergence for discretizations of the attractor

We consider the discretization of a dynamical system given bya C⁰-semigroup S(t), defined on a Banach space X, possessingan attractor . Under certain weak assumptions, Hale, Lin andRaugel showed that discretizations of S(t) possess local attractors,which may be considered as approximations to . Without furtherassumptions, we show that these local attractors possess convergentsubsequences in the Hausdorff or set metric, whose limit isa compact invariant subset of . Using a new construction, wealso consider the Kloeden and Lorenz concept of attracting setsin a Banach space, and show under mild assumptions that discretizationspossess attracting sets converging to in the Hausdorff metric. ath{at}maths.bath.ac.uk Endre.Suli{at}comlab.ox.ac.uk     

Finite Element Approximation of a Rigid Punch Indenting a Membrane

Optimal order H¹ and L error bounds are obtained for a continuouspiecewise linear finite element approximation of an obstacleproblem, where the obstacle's height as well as the contactzone, _c, are a priori unknown. The problem models the indentationof a membrane by a rigid punch. For R², given ,g R⁺ and an obstacle defined over E we consider the minimization of |v|²_1,+gµover (v, µ) H¹₀() x R subject to v+µ on E. In additionwe show under certain nondegeneracy conditions that dist (_c,^h_c)Ch ln 1/h, where ^h_c is the finite element approximation to_c. Finally we show that the resulting algebraic problem canbe solved using a projected SOR algorithm.     

Rate of convergence of numerical approximations to homoclinic bifurcation points

For x=f (x, ), x Rⁿ, R, having a hyperbolic or semihyperbolicequilibrium p(), we study the numerical approximation of parametervalues ^* at which there is an orbit homoclinic to p(). We approximate^* by solving a finite-interval boundary value problem on J=[T_–,T₊], T_–<0<T₊, with boundary conditions that sayx(T_–) and x(T₊) are in approximations to appropriate invariantmanifolds of p(). A phase condition is also necessary to makethe solution unique. Using a lemma of Xiao-Biao Lin, we improve,for certain phase conditions, existing estimates on the rateof convergence of the computed homoclinic bifurcation parametervalue , to the true value ^*. The estimates we obtain agree withthe rates of convergence observed in numerical experiments.Unfortunately, the phase condition most commonly used in numericalwork is not covered by our results.     

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case

^** Email: Paul.Houston{at}mcs.le.ac.uk^*** Email: Janice.Robson{at}comlab.ox.ac.uk^**** Email: Endre.Suli{at}comlab.ox.ac.uk We develop a one-parameter family of hp-version discontinuousGalerkin finite element methods, parameterised by [–1,1], for the numerical solution of quasilinear elliptic equationsin divergence form on a bounded open set ^d, d 2. In particular,we consider the analysis of the family for the equation –·{µ(x, |u|)u} = f(x) subject to mixed Dirichlet–Neumannboundary conditions on . It is assumed that µ is a real-valuedfunction, µ C( x [0, )), and thereexist positive constants m_µ and M_µ such that m_µ(t– s) µ(x, t)t – µ(x, s)s M_µ(t– s) for t s 0 and all x . Using a result from the theory of monotone operators for any valueof [–1, 1], the corresponding method is shown to havea unique solution u_DG in the finite element space. If u C¹() H^k(), k 2, then with discontinuous piecewise polynomials ofdegree p 1, the error between u and u_DG, measured in the brokenH¹()-norm, is (h^s–1/p^k–3/2), where 1 s min {p+ 1, k}.     

The Convergence of Operator Approximations at Turning Points

Present address: Department of Mathematics, University of Reading, Reading RG6 2AX. We consider the convergence of solution curves of approximationsto parameter-dependent operator equations of the form G(, x)= 0. Provided G_x(, x) remains non-singular this problem is cateredfor by a simple extension to standard theory. In this paper,however, attention is concentrated on solution curves throughcertain singular points (⁰, x⁰), and the main result is thatconvergence depends on consistency and stability results forthe linear eigenvalue problem G_x(⁰, x⁰)0 = 0.     

The Method of Fundamental Solutions for the Solution of Nonlinear Plane Potential Problems

The method of fundamental solutions is described for the solutionof elliptic boundary value problems governed by Laplace's equationin the plane subject to nonlinear radiation-type boundary conditions.The effectiveness of the method is demonstrated by examiningits performance on two problems from the literature, and comparisonsare made with published results obtained using boundary elementmethods. Portions of this work were conducted while this author wasa visiting assistant professor at the University of Kentucky. Partially supported by the National Science Foundation undergrant MCS-8303287 and grant RII-8610671, and by the Commonwealthof Kentucky through the Kentucky EPSCoR Program.