I will discuss recent extensions of finite element methods allowing for general meshes and their use within adaptive algorithms.As motivating examples, I will present applications relevant to the biosciences with strong localisation of solution features, eg. local blow-up [1], and/or internal curved interfaces, eg. modelling semi-permeable membranes [2].A particularly flexible approach is the hp-version Discontinuous Galerkin (hp-dG) method. The a priori error analysis of hp-dG is already very well developed for general classes of PDEs under extremely general mesh assumptions [3] and it naturally permits local mesh and order adaptivity. However, deriving robust error estimators allowing for curved/degenerating mesh interfaces as well as developing adaptive algorithms able to exploit such flexibility is far from trivial. I will review the theory and practice of hp-dG on general meshes highlighting advantages and difficulties also in relation to other frameworks, such as the Virtual Element Method [4,5,6].

- [1] Cangiani, A., Georgoulis E.H., Kyza I., and Metcalfe S. Adaptivity and blow-up detection for nonlinear evolution problems. SISC 36, 6 (2016), A3833-A3856.
- [2] Cangiani, A., Georgoulis, E. H., and Sabawi, Y. A. Adaptive discontinuous Galerkin methods for elliptic interface problems. Math. Comp. 87 (2018), 2675–2707.
- [3] Cangiani, A., Dong, Z., Georgoulis, E.H., Houston, P. hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes. SpringerBriefs in Mathematics (2017).
- [4] Beirao da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D., and Russo, A. Basic principles of virtual element methods. M3AS 23, 4 (2013), 199-214.
- [5] Cangiani, A., Georgoulis, E. H., Pryer, T., and Sutton, O. J. A posteriori error estimates for the virtual element method. Numer. Math. 137, 4 (2017), 857-893.
- [6] Brenner, S. C. and Sung, L.-Y. Virtual element Methods on meshes with small edges or faces. M3AS (2018).