Norbert Heuer, Pontificia Universidad Católica de Chile

In recent years the DPG method has raised some attention. It is a discontinuous Petrov-Galerkin method where the selection of special test functions guarantees discrete stability. In this way, for a given well-posed problem, any well-posed variational formulation is appropriate to set up a Galerkin approximation.

Practical and theoretical reasons suggest to use ultraweak variational formulations. In this case, field variables are considered in L2 so that test functions carry all the appearing derivatives. Transferring derivatives to test functions by integrating by parts, this gives rise to trace terms and thus, trace operators. In the ultraweak case, trace operators carry all the regularity weight of the problem. They have to be defined in appropriate spaces with corresponding images. They also carry the burden of conformity, when and where wanted.

Independently of the ultraweak formulation and implied DPG scheme, the conformity of trace approximations is essential to understand and characterize the conformity of Galerkin schemes in general. We discuss this relation, and strategies and arising difficulties of this approach in the case of fourth-order operators.