Pratyuksh Bansal | Searcharxiv

arXiv2021

Numerical approximation of statistical solutions of the incompressible Navier-Stokes Equations

Statistical solutions, which are time-parameterized probability measures on spaces of square-integrable functions, have been established as a suitable framework for global solutions of incompressible Navier-Stokes equations (NSE). We compute numerical approximations of statistical solutions of NSE on two-dimensional domains with non-periodic boundary conditions and empirically investigate the convergence of these approximations and their observables. For the numerical solver, we use Monte Carlo sampling with an H(div)-FEM based deterministic solver. Our numerical experiments for high Reynolds number turbulent flows demonstrate that the statistics and observables of the approximations converge. We also develop a novel algorithm to compute structure functions on unstructured meshes.

math.NA↗

arXiv2020

Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal domains $Ω\subset\mathbb{R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space-time DG formulation developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for the space-time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable \emph{sparse} space-time version of the DG scheme. The latter scheme is based on the so-called \emph{combination formula}, in conjunction with a family of anisotropic space-time DG-discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $Ω$ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space-time DG schemes.

math.NA↗