Pavel B. Bochev

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We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems,(More)
Least-squares finite element methods are an attractive class of methods for the numerical solution of partial differential equations. They are motivated by the desire to recover, in general settings, the advantageous features of Rayleigh–Ritz methods such as the avoidance of discrete compatibility conditions and the production of symmetric and positive(More)
We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB(More)
A new stabilized finite element method for the Stokes problem is presented. The method is obtained by modification of the mixed variational equation by using local L polynomial pressure projections. Our stabilization approach is motivated by the inherent inconsistency of equal-order approximations for the Stokes equations, which leads to an unstable mixed(More)
problem (Girault, Raviart, 1984) F (λ, U) ≡ U + T ·G(λ, U) = 0 , where Λ ⊂ R is compact interval; X and Y are Banach<lb>spaces and T ∈ L(Y,X). Regular branch of solutions<lb>Assume that {(λ, U(λ) |λ ∈ Λ} is such that<lb>F (λ, U(λ)) = 0 for λ ∈ Λ . 1. The set {(λ, U(λ) |λ ∈ Λ} is called branch of solu-<lb>tions if the map λ → U(λ) is a continuous(More)
This paper develops a least-squares approach to the solution of the incompressible Navier–Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier–Stokes equations as a first-order system by introducing a velocity-flux variable and associated curl and trace equations. We show that a least-squares principle(More)
Proliferation of degrees-of-freedom has plagued discontinuous Galerkin methodology from its inception over 30 years ago. This paper develops a new computational formulation that combines the advantages of discontinuous Galerkin methods with the data structure of their continuous Galerkin counterparts. The new method uses local, element-wise problems to(More)
We develop and analyze a numerical method for creating an adaptive moving grid in one, two and three-dimensional regions. The method distributes grid nodes according to a given analytic or discrete weight function of the spatial and time variables, which reflects the fine structure of the solution. The weight function defines a vector field which is used to(More)
In this paper we consider the application of least-squares principles to the approximate solution of the Stokes equations cast into a first-order velocity-vorticity-pressure system. Among the most attractive features of the resulting methods are that the choice of approximating spaces is not subject to the LBB condition and a single continuous piecewise(More)