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.     

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.     

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.     

Optimal Recovery in the Finite-Element Method, Part 1: Recovery from Weighted L2 Fits

Finite-element methods lead to approximations that are optimalor near-optimal in an associated energy norm. The consequentialproblem of recovering point values of the solution or its derivativesis addressed in this paper and its companion. A general frameworkis adopted, based on the seminal paper of Golomb & Weinberger(1959), which brings together several ideas such as superconvergence,local averaging and defect correction. Detailed results aregiven for piecewise constant and 2 linear approximations inone dimension: recovery from weighted L² fits is treated here,and from weighted H¹ fits in Part II     

Optimal Petrov--Galerkin Methods through Approximate Symmetrization

^*Present address: Department of Mathematics, Imperial College, London SW7 2BZ. A technique of approximate symmetrization is used to derivea test space from a given trial space for a Petrov—Galerkinmethod. This is applied to one-dimensional diffusion—convectionproblems to give approximations which are near optimal in anenergy norm. Rigorous and precise error bounds are derived todemonstrate the uniformly good behaviour and near optimalityof the procedure over all values of the mesh Péclet number.     

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

An initial-boundary-value problem for a parabolic equation ina domain x (0, T) with prescribed Dirichlet data on is approximatedusing a continuous-time Galerkin finite-element scheme. It isshown that the total flux across ₁= can be approximated withan error of O(h^k) when is a curved domain in Rⁿ (n = 2 or 3)and isoparametric elements having approximation power h^k inthe L² norm are used.     

A Finite-element Method for Solving Elliptic Equations with Neumann Data on a Curved Boundary Using Unfitted Meshes

This paper considers a finite-element approximation of a Poissonequation in a region with a curved boundary on which a Neumanncondition is prescribed. Piecewise linear and bilinear elementsare used on unfitted meshes with the region of integration beingreplaced by a polygonal approximation. It is shown, despitethe variational crimes, that the rate of convergence is stillorder (h) in the H¹ norm. Numerical examples show that the methodis easy to implement and that the predicted rate of convergenceis obtained. Supported by SERC postdoctoral fellowship RF/5830.     

On the Convergence of Sequences of Rational Approximations to Analytic Functions of a Certain Class

Complex analytic methods based on the theory of Walsh (1935)and on properties of orthonormal polynomials with general weight-functionon [–1, 1] are applied to the construction of variousrational approximations on the interval [–k, k] to a functiong(x) defined by or by where Remainder estimates are obtained, and from these, in the caseswhere g(x) is real on [–k, k], an asymptotic formula isobtained for the maximum error of the best rational approximationin the sense of the uniform norm. It is also shown that therate of convergence of the sequence of best approximations ofdegree n is twice the minimum rate predicted by Walsh's theory,in the sense that the degree of the approximation required fora given precision is approximately only half as great. Graphs are shown which illustrate that, for a simple example,the remainder estimates on which this asymptotic formula isbased are remarkably accurate even for approximations of lowdegree.     

Momentum-dominated orifice flow in hydraulic buffers

A study has been made of the orifice flow that occurs in a hydraulicbuffer. A technique was needed to explain phenomena that wereoccurring in some extreme situations and to provide an estimatefor parameters in a dynamic simulation model of the buffer.It has been shown that finite-element calculations of steady-statelaminar flow, with mass-conserving elements and streamline upwinding,can be used to estimate the flow within the orifice. In particular,the technique is sufficient to predict a cavitation bubble justbeyond the orifice entrance. An iterative inviscid potentialflow has been utilized to redesign the orifice boundary shapeto give a more streamlined flow. This model also confirmed thatthe ‘vena contracta’ parameter was close to thevalue of 0.9 which is needed for dynamic simulation models.Experiments confirm that the method developed suggests new orificedesigns with significantly improved performance.