# Pavel B. Bochev

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• SIAM Review
• 1998
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)
• Applied mathematical sciences
• 2009
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)
• SIAM J. Numerical Analysis
• 2006
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)
Compatible discretizations transform partial differential equations to discrete algebraic problems that mimic fundamental properties of the continuum equations. We provide a common framework for mimetic discretizations using algebraic topology to guide our analysis. The framework and all attendant discrete structures are put together by using two basic(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)