53,499 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
A Parallel Method for Population Balance Equations Based on the Method of Characteristics
In this paper, we present a parallel scheme to solve the population balance
equations based on the method of characteristics and the finite element
discretization. The application of the method of characteristics transform the
higher dimensional population balance equation into a series of lower
dimensional convection-diffusion-reaction equations which can be solved in a
parallel way.Some numerical results are presented to show the accuracy and
efficiency.Comment: 10 pages, 0 figur
Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations
Error estimates with optimal convergence orders are proved for a stabilized
Lagrange-Galerkin scheme for the Navier-Stokes equations. The scheme is a
combination of Lagrange-Galerkin method and Brezzi-Pitkaranta's stabilization
method. It maintains the advantages of both methods; (i) It is robust for
convection-dominated problems and the system of linear equations to be solved
is symmetric. (ii) Since the P1 finite element is employed for both velocity
and pressure, the number of degrees of freedom is much smaller than that of
other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is
efficient especially for three-dimensional problems. The theoretical
convergence orders are recognized numerically by two- and three-dimensional
computations
Stabilised finite element methods for ill-posed problems with conditional stability
In this paper we discuss the adjoint stabilised finite element method
introduced in, E. Burman, Stabilized finite element methods for nonsymmetric,
noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on
Scientific Computing, and how it may be used for the computation of solutions
to problems for which the standard stability theory given by the Lax-Milgram
Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to
ill-posed problems that have some conditional stability property and prove
(conditional) error estimates in an abstract framework. As a model problem we
consider the elliptic Cauchy problem and provide a complete numerical analysis
for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building
Bridges: Connections and Challenges in Modern Approaches to Numerical Partial
Differential Equation
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