15,118 research outputs found
Solving the incompressible surface Navier-Stokes equation by surface finite elements
We consider a numerical approach for the incompressible surface Navier-Stokes
equation on surfaces with arbitrary genus . The approach is
based on a reformulation of the equation in Cartesian coordinates of the
embedding , penalization of the normal component, a Chorin
projection method and discretization in space by surface finite elements for
each component. The approach thus requires only standard ingredients which most
finite element implementations can offer. We compare computational results with
discrete exterior calculus (DEC) simulations on a torus and demonstrate the
interplay of the flow field with the topology by showing realizations of the
Poincar\'e-Hopf theorem on -tori
Stabilized nonconforming finite element methods for data assimilation in incompressible flows
We consider a stabilized nonconforming finite element method for data
assimilation in incompressible flow subject to the Stokes' equations. The
method uses a primal dual structure that allows for the inclusion of
nonstandard data. Error estimates are obtained that are optimal compared to the
conditional stability of the ill-posed data assimilation problem
A Trace Finite Element Method for Vector-Laplacians on Surfaces
We consider a vector-Laplace problem posed on a 2D surface embedded in a 3D
domain, which results from the modeling of surface fluids based on exterior
Cartesian differential operators. The main topic of this paper is the
development and analysis of a finite element method for the discretization of
this surface partial differential equation. We apply the trace finite element
technique, in which finite element spaces on a background shape-regular
tetrahedral mesh that is surface-independent are used for discretization. In
order to satisfy the constraint that the solution vector field is tangential to
the surface we introduce a Lagrange multiplier. We show well-posedness of the
resulting saddle point formulation. A discrete variant of this formulation is
introduced which contains suitable stabilization terms and is based on trace
finite element spaces. For this method we derive optimal discretization error
bounds. Furthermore algebraic properties of the resulting discrete saddle point
problem are studied. In particular an optimal Schur complement preconditioner
is proposed. Results of a numerical experiment are included
Error estimates for a numerical approximation to the compressible barotropic Navier-Stokes equations
We present here a general method based on the investigation of the relative
energy of the system, that provides an unconditional error estimate for the
approximate solution of the barotropic Navier Stokes equations obtained by time
and space discretization. We use this methodology to derive an error estimate
for a specific DG/finite element scheme for which the convergence was proved in
[27]. This is an extended version of the paper submitted to IMAJNA
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