Comparison of an h- and hp-adaptive finite element solver for chemo-mechanically coupled battery electrode particles

Numerical investigations of mechanical stresses for phase transforming battery electrode materials on the particle scale are computationally highly demanding. The limitations are mainly induced by the strongly varying spatial and temporal scales of the underlying phase field model, which require an ultra fine mesh and time resolution, however, solely at specific stages in space and time. To overcome these numerical difficulties we present a general-purpose space and time adaptive solution algorithm based on an $hp$-adaptive finite element method and a variable-step, variable-order time integrator. At the example of a chemo-mechanical electrode particle model we demonstrate the computational savings gained by the $hp$-adaptivity. In particular, we compare the results to an $h$-adaptive finite element method and show the reduction of computational complexity.