1,279 research outputs found
Robust error estimates in weak norms for advection dominated transport problems with rough data
We consider mixing problems in the form of transient convection--diffusion
equations with a velocity vector field with multiscale character and rough
data. We assume that the velocity field has two scales, a coarse scale with
slow spatial variation, which is responsible for advective transport and a fine
scale with small amplitude that contributes to the mixing. For this problem we
consider the estimation of filtered error quantities for solutions computed
using a finite element method with symmetric stabilization. A posteriori error
estimates and a priori error estimates are derived using the multiscale
decomposition of the advective velocity to improve stability. All estimates are
independent both of the P\'eclet number and of the regularity of the exact
solution
Breaking spaces and forms for the DPG method and applications including Maxwell equations
Discontinuous Petrov Galerkin (DPG) methods are made easily implementable
using `broken' test spaces, i.e., spaces of functions with no continuity
constraints across mesh element interfaces. Broken spaces derivable from a
standard exact sequence of first order (unbroken) Sobolev spaces are of
particular interest. A characterization of interface spaces that connect the
broken spaces to their unbroken counterparts is provided. Stability of certain
formulations using the broken spaces can be derived from the stability of
analogues that use unbroken spaces. This technique is used to provide a
complete error analysis of DPG methods for Maxwell equations with perfect
electric boundary conditions. The technique also permits considerable
simplifications of previous analyses of DPG methods for other equations.
Reliability and efficiency estimates for an error indicator also follow.
Finally, the equivalence of stability for various formulations of the same
Maxwell problem is proved, including the strong form, the ultraweak form, and a
spectrum of forms in between
Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index
We consider a system of nonlinear partial differential equations describing
the motion of an incompressible chemically reacting generalized Newtonian fluid
in three space dimensions. The governing system consists of a steady
convection-diffusion equation for the concentration and a generalized steady
power-law-type fluid flow model for the velocity and the pressure, where the
viscosity depends on both the shear-rate and the concentration through a
concentration-dependent power-law index. The aim of the paper is to perform a
mathematical analysis of a finite element approximation of this model. We
formulate a regularization of the model by introducing an additional term in
the conservation-of-momentum equation and construct a finite element
approximation of the regularized system. We show the convergence of the finite
element method to a weak solution of the regularized model and prove that weak
solutions of the regularized problem converge to a weak solution of the
original problem.Comment: arXiv admin note: text overlap with arXiv:1703.0476
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