SEMINAIRE D'ANALYSE NUMERIQUE
Année universitaire 2012-2013
Jeudi 6 juin 2013 :
Leszek DEMKOWICZ
(ICES, The University of Texas at Austin)
Discontinuous Petrov Galerkin Method (DPG) with Optimal Test Functions
I will give a short tutorial on the DPG method proposed by Jay Gopalakrishnan and myself
four years ago, emphasizing the main points and illustrating them with numerical examples.
Here is a few of them:
1/ The DPG method is a minimum-residual method with the residual evaluated
in a dual norm.
2/ The method can be interpreted as a Petrov-Galerkin method with optimal
test functions (realizing the sup in the inf-sup condition).
3/ The optimal test functions are computed on the fly by inverting
(approximately) the Riesz operator corresponding to the test space.
4/ With broken test spaces and localizable norms, the inversion is done
elementwise, i.e. the optimal test functions are computed within
the element routine. This is more expensive then for standard FE method
but it is compatible with the standard FE technology.
5/ The main price paid for the localization is the presence of additional
unknowns: traces and fluxes. Compared with standard conforming FE methods
or hybridizable DG methods, the number of (non-local) unknowns doubles
and it is of the same range as for DG methods. Contrary to DG methods
based on numerical flux, in the DPG method, the flux enters as additional
unknown.
6/ The method can be interpreted as a preconditioned least squares method.
The stiffness matrix is hermitian and positive-definite. Its condition
number is the same as for standard FEs.
7/ The formulation based on a first order system is very natural but not
necessary. You can work with the second order equation if you wish. The
key point is to break the test functions.
8/ There is nothing exotic about the ultra-weak variational formulation
behind the DPG method. If the operator is well posed in the L^2 sense
(the operator is L^2 bounded below), the ultra-weak variational formulation
is also well posed with the corresponding inf-sup constant being of the
same order.
9/ With the use of optimal test functions, the issues of approximability
and stability are fully separated. This is illustrated by using
hp-adaptivity.
9/ The method is especially suited for singular perturbation problems e.g.
convection-dominated diffusion, high wave number wave propagation,
elasticity for thin-walled structures etc. For problems of this
type, one can systematically design a test norm to accomplish
robustness, i.e. a stability uniform in the perturbation parameter.
10/ If you have a hybrid FE code, converting it to a DPG code is very easy.
11/ The methodology extends to nonlinear problems. I will show examples
for compressible NS equations.
