S. S. Chow

Finite element error estimates for non-linear elliptic equations of monotone type

SummaryIn this paper we shall consider the application of the finite element method to a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient, and the derivation of error estimates for the finite element approximations. Such problems arise in many practical situations — for example, in shock-free airfoil design, seepage through coarse grained porous media, and in some glaciological problems. By making use of certain properties of the nonlinear coefficients, we shall demonstrate that the variational formulations associated with these boundary value problems are well-posed. We shall also prove that the abstract operators accompanying such problems satisfy certain continuity and monotonicity inequalities. With the aid of these inequalities and some standard results from approximation theory, we show how one may derive error estimates for the finite element approximations in the energy norm.

Approximate boundary-flux calculations☆

Abstract A technique for determining derivatives (fluxes or stresses) from finite element solutions is developed. The approach is a generalization to higher dimensions of a procedure known to give highly accurate results in one dimension. Numerical experiments demonstrate that certain difficulties are associated with corners in the higher-dimensional extensions and two variants of the method are examined. We consider both triangular and quadrilateral elements and observe some interesting differences in the numerical rates of convergence. Finally, this post-processing scheme is tested for nonlinear problems.

Superconvergence analysis of approximate boundary-flux calculations

SummaryCertain projection post-processing techniques have been proposed for computing the boundary flux for two-dimensional problems (e.g., see Carey, et al. [5]). In a series of numerical experiments on elliptic problems they observed that these post-processing formulas for approximate fluxes were almost (O(h2)-accurate for linear triangular elements. In this paper we prove that the computed boundary flux isO(h2 ln 1/h)-accurate in the maximum norm for the partial method of [5]. If the solutionuφH3(Ω) then the boundary flux error isO(h3/2) in theL2-norm.

On the multi-frequency inverse source problem in heterogeneous media

The inverse source problem where an unknown source is to be identified from knowledge of its radiated wave is studied. The focus is placed on the effect that multi-frequency data have on establishing uniqueness. In particular, it is shown that data obtained from finitely many frequencies are not sufficient. On the other hand, if the frequency varies within a set with an accumulation point, then the source is determined uniquely, even in the presence of highly heterogeneous media. In addition, an algorithm for the reconstruction of the source using multi-frequency data is proposed. The algorithm, based on a subspace projection method, approximates the minimum-norm solution given the available multi-frequency measurements. A few numerical examples are presented.

Determination of the transmissivity zonation using a linear functional strategy

For aquifers having a zonation structure with the transmissivity varying smoothly and slowly over each zone, a common approach in determining transmissivity is to presume a known zonation structure and seek a constant approximation to the transmissivity over each zone. However, this procedure is not always acceptable as it may lead to instability in the estimation process. By adapting the linear functional strategy proposed by Anderssen and Dietrich (1987), the authors show how one may simultaneously determine the zonation structure and a piecewise constant representation of the transmissivity. Stability results are derived for this procedure under conditions which guarantee identifiability. A generalization of this method using a Petrov-Galerkin interpretation is also examined.

Finite element error estimates for a blast furnace gas flow problem

In the mathematical modeling of gas flow in an iron-making blast furnace, a nonlinear second-order boundary value problem is obtained. The operator associated with this problem is monotone and continuous. This paper shows how the algebraic properties of the principal nonlinear coefficient may be exploited to derive optimal order energy error estimates for the finite element approximations of this problem. The situation where the coefficient is implicitly defined is also considered.

Superconvergence analysis of flux computations for nonlinear problems

In this paper we consider the error estimates for some boundary-flux calculation procedures applied to two-point semilinear and strongly nonlinear elliptic boundary value problems. The case of semilinear parabolic problems is also studied. We prove that the computed flux is superconvergent with second and third order of convergence for linear and quadratic elements respectively. Corresponding estimates for higher order elements may also be obtained by following the general line of argument. In the application of finite element methods to certain Dirichlet boundary value problems, one is often interested in estimating the flux on the boundary in addition to obtaining an approximation to the solution of the boundary value problem. A simple but effective post-processing procedure was proposed by Wheeler [18] for computing the flux from the finite element approximation of a two-point boundary value problem. By multiplying the governing equation of interest by a test function that does not vanish at both boundary points and performing an integration by parts, an equation for the flux is obtained. The finite element solution may then be used to replace the weak solution in the equation. With an appropriate choice of test function, one obtains a formula for computing the flux at the boundary points. y As it turns out, this procedure yields boundary flux values that are superconvergent, that is the error of the approximation flux has an order of convergence that is higher than that of the gradient of the finite element approximations. This observation was shown to be true rigorously for linear problems by Wheeler [20] and Dupont [10]. Extension to the two dimensional case for a rectangular domain was proposed and analysed (again for linear problems) by Douglas, Dupont and Wheeler [9]. A more computationally efficient approach was proposed by Carey [4], The crucial idea being the choice of a localised test function in the flux formula. This allows generalisation to two dimensional problems without restriction to rectangular domains.

Equivalence of \(n\)-point Gauss–Chebyshev rule and \(4n\)-point midpoint rule in computing the period of a Lotka–Volterra system

Two integrals (3.6), (4.7) for the period of a periodic solution of the Lotka–Volterra system are presented in terms of two inverse functions of

Well Singularities in Reservoir Simulation

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Modelling error and constitutive relations in simulation of flow and transport

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