Macro Elements

esys.finley supports the usage of macro elements  which can be used to achieve LBB compliance when solving incompressible fluid flow problems. LBB compliance is required to get a problem which has a unique solution for pressure and velocity. For macro elements the pressure and velocity are approximated by a polynomial of order 1 but the velocity approximation bases on a refinement of the element. The nodes of a triangle and quadrilateral element is shown in Figures 7.6(a) and 7.6(b), respectively. In essence, the velocity uses the same nodes like a quadratic polynomial approximation but replaces the quadratic polynomial by piecewise linear polynomials. In fact, this is the way esys.finley is defining the macro elements. In particular esys.finley uses the same local ordering of the nodes for the macro element as for the corresponding quadratic element. Another interpretation is that one uses a linear approximation of the velocity together with a linear approximation of the pressure but on elements created by combining elements to macro elements. Notice that the macro elements still use quadratic interpolation to represent the element and domain boundary. However, if elements have linear boundary a macro element approximation for the velocity is equivalent to using a linear approximation on a mesh which is created through a one step, global refinement. Typically macro elements are only required to use when an incompressible fluid flow problem is solved, e.g the Stokes problem in Section 6.1. Please see Section 7.2 for more details on the supported macro elements.

Table: Solvers available for esys.finley and the PASO package and the relevant options in SolverOptions. MKL supports SolverOptions.MINIMUMFILLIN and SolverOptions.NESTEDDISSECTION reordering. Currently the UMFPACK interface does not support any reordering.

setSolverMethod	DIRECT	PCG	GMRES	TFQMR	MINRES	PRES20	BICGSTAB	LUMPING
setReordering	$\checkmark$
setRestart			$\checkmark$			$20$
setTruncation			$\checkmark$			$5$
setIterMax		$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$
setTolerance		$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$
setAbsoluteTolerance		$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$
setReordering	$\checkmark$

Table: Preconditioners available for esys.finley and the PASO package and the relevant options in SolverOptions.

setPreconditioner	NO_PRECONDITIONER	AMG	JACOBI	GAUSS_SEIDEL	REC_ILU	RILU	ILU0	DIRECT
status:	later	later	$\checkmark$	$\checkmark$	$\checkmark$	later	$\checkmark$	later
setCoarsening		$\checkmark$
setLevelMax		$\checkmark$
setCoarseningThreshold		$\checkmark$
setMinCoarseMatrixSize		$\checkmark$
setNumSweeps			$\checkmark$	$\checkmark$
setNumPreSweeps		$\checkmark$
setNumPostSweeps		$\checkmark$
setInnerTolerance
setDropTolerance
setDropStorage
setRelaxationFactor						$\checkmark$
adaptInnerTolerance
setInnerIterMax