Trace operators of the bi-Laplacian and applications
This work provides theoretical foundations for ultraweak formulations of the bi-Laplace equation, relevant for numerical methods like DPG, but is incremental as it builds on prior Kirchhoff-Love traces.
The authors study trace operators and spaces for the bi-Laplacian to develop well-posed ultraweak formulations with low regularity under L2 right-hand side, achieving two well-posed formulations validated by numerical experiments with the DPG method in 2D and 3D.
We study several trace operators and spaces that are related to the bi-Laplacian. They are motivated by the development of ultraweak formulations for the bi-Laplace equation with homogeneous Dirichlet condition, but are also relevant to describe conformity of mixed approximations. Our aim is to have well-posed (ultraweak) formulations that assume low regularity, under the condition of an $L_2$ right-hand side function. We pursue two ways of defining traces and corresponding integration-by-parts formulas. In one case one obtains a non-closed space. This can be fixed by switching to the Kirchhoff-Love traces from [Führer, Heuer, Niemi, An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation, Math. Comp., 88 (2019)]. Using different combinations of trace operators we obtain two well-posed formulations. For both of them we report on numerical experiments with the DPG method and optimal test functions. In this paper we consider two and three space dimensions. However, with the exception of a given counterexample in an appendix (related to the non-closedness of a trace space), our analysis applies to any space dimension larger than or equal to two.