STEFANO GIANI
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Discontinuous Galerkin Multi-Level Method

This project has been done in collaboration with Prof. Paul Houston and Dr. Paola Antonietti.

We introduced the hp-version discontinuous Galerkin composite finite element method (DGCFEM) for the discretization of second-order 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 micro-structures. 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 micro-structure 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 micro-structure 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

 Figure 4. Convergence of the error estimator and the DG norm of the error using hp-adaptivity

We also developed the a posteriori error estimation of hp-version discontinuous Galerkin composite finite element methods for the discretization of second-order elliptic partial differential equations. In Fig. 4 we have the convergence of the DG norm of the error and the error estimator using hp-adaptivity on a domain with micro-structure. As can be seen, the convergence rate seems exponential. In Fig. 5 and Fig. 6 we have two hp-adapted 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. hp-adapted CFE mesh after 8 iterations of the refinement procedure Figure 6. hp-adapted 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 goal-oriented mesh adaptation to problems posed on complicated domains which may contain a huge number of local geometrical features, or micro-structures. 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 Navier-Stokes 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 h-adapted 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. h-adapted CFE mesh after 7 iterations of the refinement procedure Figure 9. h-adapted CFE mesh after 10 iterations of the refinement procedure

References:

S. Giani, P. Houston (2014), hp-Adaptive Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains. - Numerical Methods for Partial Differential Equations 30(4), 1342-1367.

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), hp-version composite discontinuous Galerkin methods for elliptic problems on complicated domains. - SISC 35(3), A1417-A1439.