19 research outputs found
Robust globally divergence-free weak Galerkin finite element methods for natural convection problems
This paper proposes and analyzes a class of weak Galerkin (WG) finite element
methods for stationary natural convection problems in two and three dimensions.
We use piecewise polynomials of degrees k, k-1, and k(k>=1) for the velocity,
pressure, and temperature approximations in the interior of elements,
respectively, and piecewise polynomials of degrees l, k, l(l = k-1,k) for the
numerical traces of velocity, pressure and temperature on the interfaces of
elements. The methods yield globally divergence-free velocity solutions.
Well-posedness of the discrete scheme is established, optimal a priori error
estimates are derived, and an unconditionally convergent iteration algorithm is
presented. Numerical experiments confirm the theoretical results and show the
robustness of the methods with respect to Rayleigh number.Comment: 32 pages, 13 figure
A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs II: Low Regularity
In the first part of this work, we analyzed a Dirichlet boundary control
problem for an elliptic convection diffusion PDE and proposed a new
hybridizable discontinuous Galerkin (HDG) method to approximate the solution.
For the case of a 2D polygonal domain, we also proved an optimal superlinear
convergence rate for the control under certain assumptions on the domain and on
the target state. In this work, we revisit the convergence analysis without
these assumptions; in this case, the solution can have low regularity and we
use a different analysis approach. We again prove an optimal convergence rate
for the control, and present numerical results to illustrate the convergence
theory