9,038 research outputs found
Convergence and Optimality of Adaptive Mixed Finite Element Methods
The convergence and optimality of adaptive mixed finite element methods for
the Poisson equation are established in this paper. The main difficulty for
mixed finite element methods is the lack of minimization principle and thus the
failure of orthogonality. A quasi-orthogonality property is proved using the
fact that the error is orthogonal to the divergence free subspace, while the
part of the error that is not divergence free can be bounded by the data
oscillation using a discrete stability result. This discrete stability result
is also used to get a localized discrete upper bound which is crucial for the
proof of the optimality of the adaptive approximation
A numerical study of fluids with pressure dependent viscosity flowing through a rigid porous medium
In this paper we consider modifications to Darcy's equation wherein the drag
coefficient is a function of pressure, which is a realistic model for
technological applications like enhanced oil recovery and geological carbon
sequestration. We first outline the approximations behind Darcy's equation and
the modifications that we propose to Darcy's equation, and derive the governing
equations through a systematic approach using mixture theory. We then propose a
stabilized mixed finite element formulation for the modified Darcy's equation.
To solve the resulting nonlinear equations we present a solution procedure
based on the consistent Newton-Raphson method. We solve representative test
problems to illustrate the performance of the proposed stabilized formulation.
One of the objectives of this paper is also to show that the dependence of
viscosity on the pressure can have a significant effect both on the qualitative
and quantitative nature of the solution
Optimal Flood Control
A mathematical model for optimal control of the water levels in a chain of
reservoirs is studied. Some remarks regarding sensitivity with respect to the time horizon, terminal cost and forecast of inflow are made
New recurrence relationships between orthogonal polynomials which lead to new Lanczos-type algorithms
Lanczos methods for solving Ax = b consist in constructing a sequence of vectors (Xk),k = 1,... such that rk = b-AXk= Pk(A)r0, where Pk is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(Îľi) = (y, Air0). Let P(1)k be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to c(1) defined as c(1)(Îľi) = c(Îľi+1). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for Pk and one for P(1)k. We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all
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