Solutions of Navier-Stokes Equations with Coriolis Force in the Rotational Framework

In this article, for 0 ≤m<∞ and the index vectors q=(q_1,q_2 ,q_3 ),r=(r_1,r_2,r_3) where 1≤q_i≤∞,1<r_i<∞ and 1≤i≤3, we study new results of Navier-Stokes equations with Coriolis force in the rotational framework in mixed-norm Sobolev-Lorentz spaces H ̇^(m,r,q) (R^3), which are more general than the classical Sobolev spaces. We prove the existence and uniqueness of solutions to the Navier-Stokes equations (NSE) under Coriolis force in the spaces L^∞([0, T]; H ̇^(m,r,q) ) by using topological arguments, the fixed point argument and interpolation inequalities. We have achieved new results compared to previous research in the Navier-Stokes problems.

p- and hp- virtual elements for the Stokes problem

We analyse the p- and hp-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. 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 Beirão da Veiga et al. (Numer. Math. 138(3), 581–613, 2018) 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 p 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.

A note on time-fractional Navier–Stokes equation and multi-Laplace transform decomposition method

In this study, the double Laplace Adomian decomposition method and the triple Laplace Adomian decomposition method are employed to solve one- and two-dimensional time-fractional Navier–Stokes problems, respectively. In order to examine the applicability of these methods some examples are provided. The presented results confirm that the proposed methods are very effective in the search of exact and approximate solutions for the problems. Numerical simulation is used to sketch the exact and approximate solution.

Composable, Scalable Solvers on Staggered Grids

&lt;p&gt;Scalable preconditioners for saddle point problems are essential to the solution of problems in geodynamics and beyond. Recent years have produced a wealth of research into efficient solvers for finite element methods. &amp;#160;These solvers are also effective, however, for orthogonal-grid finite volume discretizations of saddle point problems, also know as &quot;staggered grid&quot; or &quot;marker and cell (MAC)&quot; methods. Perhaps, ironically, due to the highly-structured nature of these discretizations, the use of advanced solvers is stymied due to the lack of a uniform topological abstraction, which is required for most scalable solvers, such as geometric multigrid. &amp;#160;We present new software to allow experimentation with and composition of these advanced solvers. &amp;#160;We focus on variable-viscosity Stokes problems with discontinuous coefficient jumps. &amp;#160;In particular, we attempt to demonstrate how the important know robust preconditioners may be employed, and how new variants may be experimented with. &amp;#160;Important solvers are compositions of block factorizations and multigrid cycles. &amp;#160;We demonstrate as many of these as possible, including triangular block preconditioners with nested multigrid solves, and monolithic multigrid solves with cellwise (Vanka) or field-based (Distributed Gauss-Seidel, Braess-Sarazin) smoothers. &amp;#160;Implementations are provided as part of the PETSc library, using the new DMStag component, and examples from the StagBL library are also shown where appropriate. &amp;#160;These tools are intended to help break down the barrier between cutting-edge research into advanced solvers (which is only becoming more complex, as multi-phase problems are further explored) and practical usage in geophysical research and production codes.&lt;/p&gt;