Discontinuous Galerkin framework for adaptive solution of parabolic problems

Abstract

SUMMARYNon-conforming meshes are frequently employed in adaptive analyses and simulations of multi-component systems. We develop a discontinuous Galerkin formulation for the discretization of parabolic problems that weakly enforces continuity across non-conforming mesh interfaces. A benefit of the DG scheme is that it does not introduce constraint equations and their resulting Lagrange multiplier fields as done in mixed and mortar methods. The salient features of the formulation are highlighted through an a priori analysis. When coupled with a mesh refinement scheme the DG formulation is able to accommodate multiple hanging nodes per element edge and leads to an effective adaptive framework for the analysis of interface evolution problems. We demonstrate our approach by analysing the Stefan problem of solidification.

A time‐staggered partitioned coupling algorithm for transient heat conduction

“…Any other choice of gives rise to an implicit scheme such as the unconditionally stable second-order accurate midpoint rule with = 1 2 or the unconditionally stable backward Euler scheme with = 1. In a monolithic approach as in [23], the discrete equation (20) is solved for v n+1 after which u n+1 is evaluated using (21).…”

A stabilized mixed discontinuous Galerkin formulation for double porosity/permeability model

“…One of the first successful applications of DG formulation to solve a practical problem was by [Reed and Hill, 1973], which addressed neutron transport. Over the years, DG methods have been successfully employed to solve hyperbolic PDEs [Brezzi et al, 2004;Pal et al, 2016], elliptic PDEs [Arnold et al, 2002;Barrios and Bustinzal, 2007;Cockburn et al, 2009b;Douglas and Dupont, 1976;Rivière et al, 1999;Rusten et al, 1996], parabolic PDEs [Douglas and Dupont, 1976;Kulkarni et al, 2007], coupling algorithms [Nakshatrala et al, 2009] and space-time finite elements [Abedi et al, 2006;Palaniappan et al, 2004]. Several variants of DG formulations have been developed over the years with varying merits for each variant.…”