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.     

An Error Estimate for a Finite-element Approximation of an Elliptic Variational Inequality Formulation of a Hele--Shaw Moving-boundary Problem

It is the purpose of this paper to estimate the error in thefinite-element approximation of a variational inequality formulationof a moving-boundary problem arising in Hele-Shaw flow. Linearfinite elements on a triangulation of a polygonal approximationof the domain are used. The novelty of the problem lies in thetime dependence, the semi-coercivity of the bilinear form andthe change in domain. To analyse the approximation, operatorsare introduced which split the space H1. Order h convergenceis proved in the H¹-Sobolev norm.     

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     

Computation of Flashing Flows In Variable Cross-Section Ducts

A Finite-Volume scheme which enables computing fast transient single phase and two phase flows in variable cross section ducts is presented. The physical modelling relies on the Homogeneous Relaxation Model. The theoretical framework is briefly presented, together with the main ideas of a new approximate Riemann solver. Numerical results confirm the suitability of the present approach.     

Linear hydrodynamical stability of two-layer inclined flow with viscosity and density stratifications

The linear hydrodynamical stability of two superimposed fluidsof different densities and viscosities flowing down an inclinedplane at the limit of zero Reynolds number is studied. Thereare two modes of disturbances at zero Reynolds number, one isthe free surface mode and the other is the interfacial mode.The free surface mode is shown to be always stable. The stabilityof the flow system is governed by the interfacial mode, andstability properties depend on the values of the ratios of densities,viscosities, and depths of the two fluid layers. Asymptoticproperties of stability for the limiting cases of long wavesand small depth ratio are examined and the corresponding neutralstability curves for the governing interfacial mode are provided.Growth rate curves against wavenumber for various sets of parametervalues are presented. The authors conclude with some remarkson how to obtain a more stable two-layer fluid system.