6 research outputs found
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
A Primal-Dual Weak Galerkin Method for Div-Curl Systems with low-regularity solutions
This article presents a new primal-dual weak Galerkin finite element method
for the div-curl system with tangential boundary conditions and low-regularity
assumptions on the solution. The numerical scheme is based on a weak
variational form involving no partial derivatives of the exact solution
supplemented by a dual or ajoint problem in the general context of the weak
Galerkin finite element method. Optimal order error estimates in are
established for solution vector fields in .
The mathematical theory was derived on connected domains with general
topological properties (namely, arbitrary first and second Betti numbers).
Numerical results are reported to confirm the theoretical convergence
Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem
We consider primal-dual mixed finite element methods for the solution of the elliptic
Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and
determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that
yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also
accounted for. A reduced version of the method, obtained by choosing a special stabilization of the
dual variable, can be viewed as a variant of the least squares mixed finite element method introduced
by Dard´e, Hannukainen, and Hyv¨onen in [SIAM J. Numer. Anal., 51 (2013), pp. 2123–2148]. The
main difference is that our choice of regularization does not depend on auxiliary parameters, the
mesh size being the only asymptotic parameter. Finally, we show that the reduced method can
be used for defect correction iteration to determine the solution of the full method. The theory is
illustrated by some numerical examples