Provable stable arbitrary order symmetric interior penalty discontinuous Galerkin (SIP) discretisations of variable viscosity, incompressible Stokes flow utilising @math -- @math elements and hierarchical Legendre basis polynomials are developed and investigated.For solving the resulting linear system, a block preconditioned iterative method is proposed. The nested viscous problem is solved by a @math -multilevel preconditioned Krylov subspace method. For the @math -coarsening, a twolevel method utilising element-block Jacobi preconditioned iterations as a smoother is employed. Piecewise bilinear ( @math ) and piecewise constant ( @math ) @math -coarse spaces are considered. Finally, Galerkin @math -coarsening is proposed and investigated for the two @math -coarse spaces considered. Through a number of numerical experiments, we demonstrate that utilising the @math coarse space results in the most robust @math -multigrid method for variable viscosity Stokes flow. Using this @math coarse space we observe that the convergence of the overall Stokes solver appears to be robust with respect to the jump in the viscosity and only mildly depending on the polynomial order @math . It is demonstrated and supported by theoretical results that the convergence of the SIP discretisations and the iterative methods rely on a sharp choice of the penalty parameter based on local values of the viscosity.
Inf-sup stable discontinuous Galerkin (DG) methods of the interior penalty type for the Stokes equations can be constructed by using the tensor product element pairs @math -- @math , and @math -- @math @cite_23 @cite_39 , as well as the @math -conforming Raviart-Thomas, Brezzi-Douglas-Marini, and Brezzi-Douglas-Fortin-Marini kind element pairs; see @cite_12 @cite_3 @cite_48 @cite_27 .
@cite_48 @cite_3 @cite_39 @cite_27 @cite_23 @cite_12
1963859447 2225768769 1982359839 2157023312
In this paper we construct Discontinuous Galerkin approximations of the Stokes problem where the velocity field is @math H ( div , Ω ) -conforming. This implies that the velocity solution is divergence-free in the whole domain. This property can be exploited to design a simple and effective preconditioner for the final linear system. A multigrid method for the Stokes system discretized with an Hdiv-conforming discontinuous Galerkin method is presented. It acts on the combined velocity and pressure spaces and thus does not need a Schur complement approximation. The smoothers used are of overlapping Schwarz type and employ a local Helmholtz decomposition. Additionally, we use the fact that the discretization provides nested divergence free subspaces. We present the convergence analysis and numerical evidence that convergence rates are not only independent of mesh size, but also reasonably small. We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using ℚk′–ℚk velocity-pressure pairs with k′ = k + 2, k + 1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half of the power of k is lost for p and hp pproximations independently of the divergence stability. In this paper, the authors present two formulations for the Stokes problem which make use of the existing @math elements of the Raviart-Thomas type originally developed for the second-order elliptic problems. In addition, two new @math elements are constructed and analyzed particularly for the new formulations. Optimal-order error estimates are established for the corresponding finite element solutions in various Sobolev norms. The finite element solutions feature a full satisfaction of the continuity equation when existing Raviart-Thomas-type elements are employed in the numerical scheme.