An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids

Abstract

In this article we design and analyze a class of two-level non-overlapping additive Schwarz preconditioners for the solution of the linear system of equations stemming from discontinuous Galerkin discretizations of secondorder elliptic partial differential equations on polytopic meshes. The preconditioner is based on a coarse space and a non-overlapping partition of the computational domain where local solvers are applied in parallel. In particular, the coarse space can potentially be chosen to be non-embedded with respect to the finer space; indeed it can be obtained from the fine grid by employing agglomeration and edge coarsening techniques. We investigate the dependence of the condition number of the preconditioned system with respect to the diffusion coefficient and the discretization parameters, i.e., the mesh size and the polynomial degree of the fine and coarse spaces. Numerical examples are presented which confirm the theoretical bounds.

Efficient multigrid solution of elliptic interface problems using viscosity-upwinded local discontinuous Galerkin methods

“…• The terms J h and J g collect the entire influence of the jump data g i j , h i j and boundary data g, h, including that which is incorporated in penalization in (7) and the numerical fluxes (4) and (6).…”

“…In particular, Figure 2, top row, illustrates the piecewise-cubic discrete solution and its error, showing a pronounced numerical boundary layer. 7 The condition number of the preconditioned system V A is approximately 5200; inspection of the spectrum of V A, consisting of n( p + 1) = 64 eigenvalues (see Figure 2, center left) shows that the smallest eigenvalue λ min ≈ 8.5×10 −4 is the main contributor to the poor condition number; the corresponding piecewisecubic eigenfunction is essentially identical (up to normalization) to the error profile shown in Figure 2, top right. Thus, in this particular example, the mode which contributes to poor accuracy happens to be the same mode which multigrid most ineffectively handles.…”

p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

“…The number of smoothing iterations has been experimentally selected so as to guarantee the best computational efficiency on the meshes considered in the numerical tests. Other choices for the smoothers could be considered such as, e.g., the ones proposed in the recent work [6]; we postpone the investigation of this topic to a future work. On the coarse level, we employ an LU solver when working in two space dimensions and ILU preconditioned GMRES solver when working in three space dimensions.…”

ℎ𝑝-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements

“…Finally, curved element capabilities should ideally be developed in conjunction with the already developed highly general polytopic mesh IP-dG methods, allowing for instance elements with arbitrary number of faces. This is particularly pertinent in the contexts of adaptivity and multilevel solvers, which benefit from element agglomeration [2,3] to achieve coarser representations. With regard to adaptivity, mesh coarsening is essential in keeping the computation sizes at bay, at least in the case of evolution problems.…”