121 research outputs found
A priori and a posteriori analysis of non-conforming finite elements with face penalty for advection-diffusion equations
We analyse a non-conforming finite-element method to approximate advection-diffusion-reaction equations. The method is stabilized by penalizing the jumps of the solution and those of its advective derivative across mesh interfaces. The a priori error analysis leads to (quasi-)optimal estimates in the mesh size (sub-optimal by order ½ in the L2-norm and optimal in the broken graph norm for quasi-uniform meshes) keeping the Péclet number fixed. Then, we investigate a residual a posteriori error estimator for the method. The estimator is semi-robust in the sense that it yields lower and upper bounds of the error which differ by a factor equal at most to the square root of the Péclet number. Finally, to illustrate the theory we present numerical results including adaptively generated meshe
Primal dual mixed finite element methods for indefinite advection--diffusion equations
We consider primal-dual mixed finite element methods for the
advection--diffusion equation. For the primal variable we use standard
continuous finite element space and for the flux we use the Raviart-Thomas
space. We prove optimal a priori error estimates in the energy- and the
-norms for the primal variable in the low Peclet regime. In the high
Peclet regime we also prove optimal error estimates for the primal variable in
the norm for smooth solutions. Numerically we observe that the method
eliminates the spurious oscillations close to interior layers that pollute the
solution of the standard Galerkin method when the local Peclet number is high.
This method, however, does produce spurious solutions when outflow boundary
layer presents. In the last section we propose two simple strategies to remove
such numerical artefacts caused by the outflow boundary layer and validate them
numerically.Comment: 25 pages, 6 figures, 5 table
Theoretical and numerical studies of chaotic mixing
Theoretical and numerical studies of chaotic mixing are performed to circumvent the difficulties
of efficient mixing, which come from the lack of turbulence in microfluidic devices. In order to
carry out efficient and accurate parametric studies and to identify a fully chaotic state, a spectral
element algorithm for solution of the incompressible Navier-Stokes and species transport
equations is developed. Using Taylor series expansions in time marching, the new algorithm
employs an algebraic factorization scheme on multi-dimensional staggered spectral element
grids, and extends classical conforming Galerkin formulations to nonconforming spectral
elements. Lagrangian particle tracking methods are utilized to study particle dispersion in the
mixing device using spectral element and fourth order Runge-Kutta discretizations in space and
time, respectively. Comparative studies of five different techniques commonly employed to
identify the chaotic strength and mixing efficiency in microfluidic systems are presented to
demonstrate the competitive advantages and shortcomings of each method. These are the stirring
index based on the box counting method, Poincare sections, finite time Lyapunov exponents, the
probability density function of the stretching field, and mixing index inverse, based on the
standard deviation of scalar species distribution. Series of numerical simulations are performed
by varying the Peclet number (Pe) at fixed kinematic conditions. The mixing length (lm) is characterized as function of the Pe number, and lm ∝ ln(Pe) scaling is demonstrated for fully
chaotic cases. Employing the aforementioned techniques, optimum kinematic conditions and the
actuation frequency of the stirrer that result in the highest mixing/stirring efficiency are
identified in a zeta potential patterned straight micro channel, where a continuous flow is
generated by superposition of a steady pressure driven flow and time periodic electroosmotic
flow induced by a stream-wise AC electric field. Finally, it is shown that the invariant manifold
of hyperbolic periodic point determines the geometry of fast mixing zones in oscillatory flows in
two-dimensional cavity
Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?
The contents of this paper is twofold. First, important recent results concerning finite element
methods for convection-dominated problems and incompressible flow problems are described that
illustrate the activities in these topics. Second, a number of, in our opinion, important problems in
these fields are discussed
A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem
A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variable u and the auxiliary
variable σ with respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results
to confirm the theoretical analysis
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