BAIL 2006 Book of Abstracts - Institut für Numerische und ...

A.CANGIANI, E.H.GEORGOULIS, M. JENSEN: Continuous-Discontinuous Finite
Element Methods for Convection-Diffusion Problems
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Continuous-Discontinuous Finite Element Methods for
Convection-Diffusion Problems
Abstract
Andrea Cangiani, Emmanuil H. Georgoulis and Max Jensen
February 14, 2006
Standard (conforming) finite element approximations of convection-dominated convection-diffusion
problems often exhibit poor stability properties that manifest themselves as non-physical oscillations
polluting the numerical solution. Various techniques have been proposed for the stabilisation of finite
element methods (FEMs) for convection-diffusion problems, see for example, Morton [12] and Roos,
Stynes and Tobiska [14] for a complete survey. Common techniques are Petrov-Galerkin methods,
like the streamline upwind Petrov-Galerkin (SUPG) method introduced by Hughes and Brooks [8],
exponential fitting [13], ad hoc meshing, like graded meshes [15] and Shishkin type meshes [11], and
adaptive mesh refinement (see, e.g., [5] and [6]). More recently, the residual-free bubble method of
Brezzi et al [2], [1], [4] and the variational multiscale method of Hughes and co-authors [9],[10].
During the last decade, families of discontinuous Galerkin finite element methods (DGFEMs)
have been proposed for the numerical solution of convection-diffusion problems, due to many attractive
properties they exhibit. In particular, DGFEMs admit good stability properties, they offer
flexibility in the mesh design (irregular meshes are admissible) and in the imposition of boundary
conditions (Dirichlet boundary conditions are weakly imposed), and they are increasingly popular
in the context of hp-adaptive algorithms.
The above mentioned attractive features of DGFEMs come at the price of the higher number
of degrees of freedom used. For instance, for piece-wise linear approximations in d-dimensions,
DGFEMs require 2 d times more degrees of freedom than conforming FEMs and their stabilised
variants. The relative difference in the number of degrees of freedom reduces when considering
higher order local polynomial degree in the approximations; nevertheless, the conforming FEMs
always require less degrees of freedom than their DGFEM counterparts.
The issue of the number of degrees of freedom required by DGFEMs has recently been addressed
by T.J.R. Hughes and co-workers in [7], where the Multiscale Discontinuous Galerkin (MDG) finite
element method framework is introduced. MDG uses local, element-wise problems to develop a
transformation between the degrees of freedom of the discontinuous space and a related, smaller,
continuous space. The transformation enables a direct construction of the global matrix problem
in terms of the degrees of freedom of the continuous space. It is proved that the method has the
accuracy and stability of the original DGFEM. The drawback of this approach is the computational
overhead of the solution of the local element-wise problems.
We propose a numerical scheme for linear convection-diffusion problems which couples continuous
and discontinuous Galerkin finite elements (CG-DG) in a different manner. Depending on the
local variation of the solution, the scheme locally uses either the computationally less expensive
continuous finite elements or the computationally more costly but also more stable discontinuous
Galerkin discretisation. Given that good stability properties are required near features of almost
(d − 1)-dimensional character (such as boundary and/or interior layers) discontinuous Galerkin
discretisation is only used in the neighborhoods of such features, whereas standard conforming FEM
is used away from the layers. Thus, the increased overhead from the use of DGFEMs is balanced by
their minimal use, limited to the neighborhoods of layers.
This work introduces the continuous-discontinuous Galerkin (CG-DG) finite element method,
and presents the first results in the analysis of this approach. In particular, we derive an a-priori
error analysis of the CG-DG blending technique, along with numerical experiments that evaluate
the accuracy and efficiency of the CG-DG approach in practice.
Speaker: GEORGOULIS, E.H. 75 BAIL 2006
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