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Discontinuous Galerkin MultiLevel Method 

This
project has been done in collaboration with Prof.
Paul Houston and Dr. Paola Antonietti.
We introduced the hpversion discontinuous Galerkin composite finite
element method (DGCFEM) for the discretization of secondorder elliptic partial differential equations. This class
of methods allows for the approximation of problems posed on computational domains which may
contain a huge number of local geometrical features, or microstructures. While standard numerical
methods can be devised for such problems, the computational effort may be extremely high, as
the minimal number of elements needed to represent the underlying domain can be very large. In
contrast, the minimal dimension of the underlying composite finite element space is independent of
the number of geometric features. The key idea in the construction of this latter class of methods
is that the computational domain is no longer resolved by the mesh; instead, the finite element
basis (or shape) functions are adapted to the geometric details present in the domain.
DGCFEM on complicated domains:
Figure
1. Composite finite element mesh on a complicated domain

In
Fig. 1 we have an example of a domain with a microstructure composed of 256 circular holes.
In the figure we have the fine mesh describing correctly the geometry in blu and the coarse composite finite element (CFE) mesh, to coarse to describe the geometry in black.
In this example, the CFE mesh contains only 8 elements.
In Fig. 2 and Fig. 3 the convergence rate of DGCFEM on the complicated domain in Fig. 1 is compared to a standard DG method for the same problem, but on a domain without the holes.
As can be seen the two methods converge in a very similar way, showing that the presence of the microstructure does not affect the convergence rate of DGFEM.

Figure
2.
Convergence of the method in the L2 norm


Figure
3.
Convergence of the method in the H1 norm


DGCFEM with hpadaptivity:
Figure
4. Convergence of the error estimator and the DG norm of the error using hpadaptivity

We also developed the a posteriori error estimation of hpversion discontinuous
Galerkin composite finite element methods for the discretization of secondorder elliptic partial
differential equations.
In Fig. 4 we have the convergence of the DG norm of the error and the error estimator using hpadaptivity on a domain with microstructure.
As can be seen, the convergence rate seems exponential.
In Fig. 5 and Fig. 6 we have two hpadapted meshes respectively after 8 and 17 iterations of the adapted procedure. As can be seen, the mesh has been refined accordingly to the relevance of the small holes for reducing the approximation error in the computed solution.

Figure
5.
hpadapted CFE mesh after 8 iterations of the refinement procedure


Figure
6.
hpadapted CFE mesh after 17 iterations of the refinement procedure


DGFEM for computational fluid dynamics:
Figure
7. Domain with periodically placed obstacles

We have considered the application of goaloriented mesh adaptation
to problems posed on complicated domains which may contain a huge number of local geometrical features, or microstructures. We exploit the
composite variant of the discontinuous Galerkin finite element method based
on exploiting finite element meshes consisting of arbitrarily shaped element
domains. Adaptive mesh refinement is based on constructing finite element
partitions of the domain consisting of agglomerated elements which belong
to different levels of an underlying hierarchical tree data structure. As an
example of the application of these techniques, we consider the numerical
approximation of the incompressible NavierStokes equations.
In Fig. 7 we have the considered domain with obstacles and the initial CFE mesh.
In Fig. 8 and Fig. 9 we have two hadapted meshes respectively after 7 and 10 iterations of the adapted procedure using the value of the pressure in one point as quantity to minimize.

Figure
8.
hadapted CFE mesh after 7 iterations of the refinement procedure


Figure
9.
hadapted CFE mesh after 10 iterations of the refinement procedure


References:
S. Giani, P. Houston (2014), hpAdaptive Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains.
 Numerical Methods for Partial Differential Equations 30(4), 13421367.
S. Giani, P. Houston (2013), Domain Decomposition Preconditioners for Discontinuous Galerkin Discretizations of Compressible Fluid Flows.
 Numerical Mathematics: Theory, Methods and Applications, Accepted.
P. Antonietti, S. Giani and P. Houston (2013), hpversion composite discontinuous Galerkin methods for elliptic problems on complicated domains.
 SISC 35(3), A1417A1439.

