Exponential convergence of mixed hp-DGFEM for Stokes flow in polygons

Summary.: We analyze mixed hp-discontinuous Galerkin finite element methods (DGFEM) for Stokes flow in polygonal domains. In conjunction with geometrically refined quadrilateral meshes and linearly increasing approximation orders, we prove that the hp-DGFEM leads to exponential rates of convergence for piecewise analytic solutions exhibiting singularities near corner

hp-Optimal discontinuous Galerkin methods for linear elliptic problems

The aim of this paper is to overcome the well-known lack of p-optimality in hp-version discontinuous Galerkin (DG) discretizations for the numerical approximation of linear elliptic problems. For this purpose, we shall present and analyze a class of hp-DG methods that is closely related to other DG schemes, however, combines both p-optimal jump penalty as well as lifting stabilization. We will prove that the resulting error estimates are optimal with respect to both the local element sizes and polynomial degrees

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows

In this article we develop both the a priori and a posteriori error analysis of hp–version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain $\Omega \subset R^{d}, d$ = 2,3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp–adaptive refinement algorithm

Analytic regularity for the incompressible Navier-Stokes equations in polygons

In a plane polygon $P$ with straight sides, we prove analytic regularity of the Leray-Hopf solution of the stationary, viscous, and incompressible Navier-Stokes equations. We assume small data, analytic volume force and no-slip boundary conditions. Analytic regularity is quantified in so-called countably normed, corner-weighted spaces with homogeneous norms. Implications of this analytic regularity include exponential smallness of Kolmogorov $N$-widths of solutions, exponential convergence rates of mixed $hp$-discontinuous Galerkin finite element and spectral element discretizations and of model order reduction techniques

p- and hp- virtual elements for the Stokes problem

We analyse the p- and hp-versions of the virtual element method (VEM) for the the Stokes problem on a polygonal domain. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from [7] an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method

Exponential convergence of mixed hp-DGFEM for Stokes flow in polygons

ISSN:0029-599XISSN:0945-324

Exponential convergence of mixed hp-DGFEM for stokes flow in polygons

ISSN:0029-599XISSN:0945-324