1 The new staggered spacetime finite element method
1.1 The spatial discretization
We propose a numerical scheme for the solution of the incompressible NavierStokes equations on 2D domains, following the original ideas in 2DSIUSW ; 2STINS . The spatial discretization is performed through the use of two unstructured grids: a primary triangular grid for the approximated pressure function, and a staggered edgebased quadrilateral grid (named dual grid) for the approximated velocity functions. In the framework of the Galerkin finite element methods, we define the finite element space of order for the discretized pressure using the standard nodal basis functions on the reference triangular element
, by imposing the classical Lagrange interpolation condition
over the 2D NewtonCotes quadrature nodes:(1) 
with the multiindex and the index ranges and . In this way basis functions are obtained. Analogously, we choose the basis functions on the reference square element for the dual finite element space of order . Since we want to derive a quadraturefree Arbitrary Lagrangian Eulerian implementation, we consider the square as the union of two subtriangles and and we construct the basis functions following the standard nodal approach of continuous finite elements (indeed, we get two minielements). This is an approach very similar to the finite elements used for the velocity approximation in some mixed problems, for example the Stokes problem, where mixed finite element approximations are employed in order to numerically satisfy the infsup compatibility condition. These elements are fully described in ern2004 . We build the basis functions over the nodes which lie on the subtriangle exactly like we did for , and then we extend them continuously to , in order not to have discontinuity inside the square. Viceversa, we construct the basis functions over the nodes which lie on via a transformation between and , and then we extend them continuously to .
1.2 Spacetime extension
For the time discretization, the generic spacetime element defined in the th timestep of the simulation is given by a triangular prism for the main grid, and a quadrilateral prism for the dual grid. In the Lagrangian case, these volumes are stretched in some way according to the local mesh velocity, while in the Eulerian case they are some right, not slanting in time, prisms. The temporal basis functions for polynomials of degree are defined as the Lagrange interpolation polynomials passing through the equidistant 1D NewtonCotes quadrature nodes on the reference interval
. Finally, using the tensor product, we define the basis functions on the spacetime elements as
and . Therefore, the total number of basis functions becomes and .2 The semiimplicit scheme for the incompressible NavierStokes equations
We consider the two dimensional NavierStokes equations for incompressible Newtonian fluids with homogeneous density and homogeneous dynamic viscosity coefficient , in the conservative, adimensional form:
(2)  
(3) 
where indicates the normalized fluid pressure; is the physical pressure; is the kinematic viscosity coefficient;
is the velocity vector;
and are the velocity components in the and direction, respectively; is a source term; is the flux tensor of the nonlinear convective terms. Following the same ideas in ADERNSE ; MunzDiffusionFlux , the momentum Eq. (3) can be rewritten as:(4) 
where .
2.1 The Picard’s method for the highorder accuracy in time
The Discontinuous Galerkin finite element formulation considers the integration of Equations (2) and (3) on the spacetime control volumes of the primary grid and of the dual grid, respectively. Following the ideas in casulli , 2STINS , a semiimplicit discretization is employed, which combines the simplicity of explicit methods for nonlinear hyperbolic PDE with the stability and efficiency of implicit time discretizations. In order to obtain a method with highorder accuracy in time, we use a simple Picard iteration which introduces the information of the new pressure into the viscous and convective terms, but without involving a nonlinearity in the final system to be solved. This approach is inspired by the local spacetime Galerkin predictor method proposed for the high order time discretization of schemes in Dumbser2008 ; ADERNSE . One time step of the final numerical scheme can be summarized as follows:

Initialize and using the known information from the previous timestep, or the initial conditions;

Picard iteration over :

compute using in the discretized momentum equation; then set ;

update from the pressure correction values ;


set and .
The resulting linear system for the pressure correction is very sparse thanks to the use of the staggered grid, including only four nonzero blocks per element.
3 Numerical results
Some relevant tests were executed in order to assess the computational efficiency and the accuracy of the new numerical method. Compared to the staggered spacetime DG algorithm of Tavelli and Dumbser 2STINS the new method proposed here is not only computationally more efficient thanks to its quadraturefree formulation, but also less memory consuming, since all integrals can be precomputed once and for all on a universal reference element. Moreover, thanks to the use of the piecewise basis functions for the dual finite element space, all the matrices of the Galerkin formulation can be updated, with their geometric information, at every timestep , by a cheap matrixvector product which uses the first levels of the cache memory.
TaylorGreen vortex test

[Eul.] 

[Eul.] 

[Eul.] 


62  6.26E02  4.40E02  3.95E02  2.18E02     
116  2.67E02  1.87E02  1.69E02  8.43E03  2.7  3.1 
380  4.49E03  3.57E03  2.52E03  1.17E03  3.0  3.2 
902  1.36E03  1.11E03  6.50E04  2.97E04  3.2  3.2 
CPU time 
CPU time [Eul.] 
RAM usage 
RAM usage [Eul.] 


62  7  7  49  67 
116  21  22  67  106 
380  137  178  179  311 
902  636  1059  392  709 
As a numerical test for the incompressible NavierStokes equations, we consider the unsteady TaylorGreen vortex problem. The solution of the problem is determined by:
(5)  
(6)  
(7) 
We set periodic boundary conditions for the domain , and the final time . An implicit treatment of the viscosity terms is applied, therefore the time step is given by the CFLtype restriction for only the convective operator:
where , is the minimum of the radii of the circles inscribed in the triangles (primary grid) and
is the maximum, over all the edges of the quadrilaterals (dual grid), of the maximum eigenvalue of the convective flux tensor, that is
.We have compared the results obtained by the new ALE implementation of the method, using zero velocity mesh, with the results that were obtained in 2STINS by an Eulerian implementation of the method, with fixed meshes: in Table 2 the errors in the norms for the pressure and the velocity field , together with the convergence order , for an increasing size of the primary grid (first column), are reported for the case . In Figure 1 the slope of the thirdorder method is shown for this test.
Finally, Table 4 and Figure 2 show the improved efficiency of the new algorithm (ALE, with zero mesh velocity), with respect to the Eulerian implementation: for the grid with the highest resolution, of the computational time is saved, and only of the memory is required, with respect to the Eulerian implementation.
References
 (1) V. Casulli, Semiimplicit finite difference methods for the twodimensional shallow water equations, Journal of Computational Physics, 1990, vol. 86(1), pp. 56–74.
 (2) M. Dumbser, D. Balsara, E.F. Toro and C.D. Munz, A unified framework for the construction of onestep finitevolume and discontinuous Galerkin schemes, Journal of Computational Physics, 2008, vol. 227, pp. 8209–8253.
 (3) M. Dumbser, Arbitrary high order PNPM schemes on unstructured meshes for the compressible NavierStokes equations, Computers and Fluids, 2010, vol. 39, pp. 60–76.
 (4) A. Ern and J.L. Guermond, Theory and Practice of Finite Elements, SpringerVerlag, New York, 2004.
 (5) G. Gassner, F. Lorcher and C.D. Munz, A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes, Journal of Computational Physics, 2007, vol. 224, pp. 1049–1063.
 (6) M. Tavelli and M. Dumbser, A high order semiimplicit discontinuous Galerkin method for the two dimensional shallow water equations on staggered unstructured meshes, Applied Mathematics and Computation, 2014, vol. 234, pp. 623–644.
 (7) M. Tavelli and M. Dumbser, A staggered arbitrary high order semiimplicit discontinuous Galerkin method for the two dimensional incompressible NavierStokes equations, Applied Mathematics and Computation, 2014, vol. 248, pp. 70–92.
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