Multilevel Monte Carlo for Polytopic DG Discretisations with Spatially Represented Random Coefficients

arXiv:2610.31861v1 Announce Type: new Abstract: We develop a multilevel Monte Carlo framework for coercive elliptic problems with random diffusion and reaction coefficients on polytopic meshes, with particular emphasis on the approximation and inter-level coupling of random coefficients generated from spatially represented latent fields. The main analytical contribution is a Strang-type error decomposition in a DG norm that separates the forward discretisation error from the error introduced by approximating the random coefficients. On this basis, we study two coefficient-generation and multilevel-coupling strategies, a spectral sampler based on a stationary Gaussian latent field defined on a fixed Cartesian reference grid, and an SPDE-based Mat\'ern sampler in which levels are coupled through projected white-noise loads on agglomerated hierarchies. Both fit into the same coefficient-aware analytical framework and couple adjacent levels directly through a common latent random input. The theory is supported by numerical experiments on general Voronoi and agglomerated polytopic meshes, demonstrating the predicted forward and coefficient-representation errors, decay of multilevel corrections, strong inter-level coupling, and the resulting work-accuracy behaviour relative to single-level Monte Carlo.