Summary. In 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.
The inverse source problem where an unknown source is to be identified from the knowledge of its radiated wave is studied. The focus is placed on the effect that multifrequency data has on establishing uniqueness. In particular, it is shown that data obtained from finitely many frequencies is 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 multifrequency measurements. A few numerical examples are presented.
In this thesis, we examine the order of convergence of finite element approximations to some nonlinear e l l i p t i c problems of monotone type. This is determined by establishing error estimates in the 'energy 1 norm for the approximations. The nonlinear problems we are interested in arise in many practical situations. Examples from blast furnace gas flow, magnetostatic distribution and nonlinear seepage flow are discussed in some details.In Chapter 2 we show that a class of second order boundary value problems of divergence form with gradient nonlinearity may be formulated variationally, and, in an abstract setting, as an operator equation, i f the nonlinear coefficients of the equation satisfy certain conditions. Under these conditions the operator turns out to be monotone. In Chapter 3 we introduce the concept of strong monotonicity and T-continuity and demonstrate the relation between strong monotonicity and convexity. We then prove the strong monotonicity and Holder continuity of the operator introduced in Chapter 2. The well-posedness of the boundary value problem is then shown by establishing the unique solvability and stability of the problem.In Chapter U, after showing how error estimates for finite element approximations may be obtained in an abstract setting i f the associated operator is strongly monotone and T-continuous, we proceed to derive error
Summary. Certain 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
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