Linear Galerkin vs mixed finite element 2D flow fields

Abstract

Numerical velocity fields arising from the solution of diffusion equations by the finite element (FE) and the mixed hybrid finite element (MHFE) schemes display different behaviors. In this paper we analyze the characteristics of the two different velocity fields in terms of both accuracy and mass balance properties. General theoretical findings are mostly concerned with the asymptotic behavior of the numerical schemes, i.e. they look at properties as the mesh size tends to zero. For practical applications, it is necessary to work with a fixed mesh of given size. Thus, we attempt to characterize the numerical flow field accuracy by analyzing the resulting mass balance characteristics on a fixed mesh. The comparison is carried out by using direct local mass balance evaluations and by calculating streamlines. We detail the important differences, advantages, and disadvantages of the two approaches. In particular, we show that both FE and MH are perfectly conservative (up to the residual of the linear system solution) if proper control volumes are used. MH streamlines are admissible, i.e. numerical normal fluxes across cell interfaces are continuous. Since continuity of the normal fluxes is not guaranteed by FE, the resulting streamlines are less accurate

MAST-2D diffusive model for flood prediction on domains with triangular Delaunay unstructured meshes

“…It can be shown that in the Galerkin formulation the same flux is computed as [42,43]: Diffusive problem (11) and (12) is solved using the same spatial discretization adopted in the prediction problem, as well as a fully implicit time discretization. Integration of Eq.…”

“…(11) according to the standard Galerkin approach, are proportional to a parameter T assumed constant inside each element m and equal to [42,43]:…”

“…Table 2 shows the time step size Dt, the maximum CFL value calculated according to Eq. (43) in the domain during the entire simulation, the mean CPU time per computational node and per iteration and the relative error e of the solution at the peak time, between the 1st and mth simulation, calculated as:…”

“…(22b)), is: rw m s is a constant vector orthogonal to side opposite to node s, with module 1/a s (a s is the triangle height measured from node s to the opposite side (see Fig. C.1)), that can be written as [43] …”

Note on a 2DH finite element model of tidal flow of the North Passage of the partially-mixed Changjiang River estuary, China

“…The finite element has been applied in plane stress analysis (Clough, 1960); the solution of field problems (Zienkiewicz and Cheung, 1965); the geology (Cheng, 1972a, b;Cheng and Hodge, 1976); lake circulation (Cheng, 1972a); and fluid dynamics (Baker, 1975;Shen, 1977;Putti and Sartoretto, 2009). Since the finite element method (FEM) can deal with the geometric flexibility of the bathymetrical topography, the finite element method has also been utilized to model tidal flow in estuarine and coastal waters (e.g., Taylor and Davis, 1975;Kawahara and Hasegawa, 1978;Reichard and Celikkol, 1978;Walters and Cheng, 1979;Holz, 1980;Kawahara et al, 1981;Yu and Lee, 1984;Tan and Zhao, 1988;Leclerc et al, 1990;Tabuenca et al, 1992;Li and Zhan, 1993;Knock and Ryrie, 1994;Kawahara and Ding, 1998;Wai et al, 1998;Heniche et al, 2000;Shi et al, 2003;and Jiang and Wai, 2005).…”