We review recent results [10, 9, 8] on time-discrete discontinuous Ga- lerkin (dG) methods for advection-diffusion model problems defined on deformable do- mains and written on the Arbitrary Lagrangian Eulerian (ALE) framework. ALE for- mulations deal with PDEs on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity. We describe the construction of higher order in time numerical schemes enjoying stability properties independent of the arbitrary extension chosen. Our approach is based on the validity of Reynolds’ identity for dG methods which generalize to higher order schemes the Geometric Conservation Law (GCL) condition. Stability, a priori and a posteriori error analyses are briefly discussed and illustrated by insightful numerical experiments.
|Title of host publication||Recent developments in discontinuous Galerkin finite element methods for partial differential equations|
|Editors||Xiaobing Feng, Ohannes Karakashian , Yulong Xing|
|Publisher||Springer International Publishing|
|Number of pages||36|
|Publication status||Published - 2014|
|Name||IMA volumes in mathematics and its applications|
Bonito, A., Kyza, I., & Nochetto, R. H. (2014). A DG approach to higher order ALE formulations in time. In X. Feng, O. Karakashian , & Y. Xing (Eds.), Recent developments in discontinuous Galerkin finite element methods for partial differential equations (pp. 223-258). (IMA volumes in mathematics and its applications; Vol. 157). Springer International Publishing. https://doi.org/10.1007/978-3-319-01818-8_10