Page 255 - Process Modelling and Simulation With Finite Element Methods

242 Process Modelling and Sirnulation with Finite Element Methods

6.5 End Note: Solver Parameters for Problems with Pointwise Weak Terms
FEMLAB errors message(s) that inspired this note
Index into matrix is negative or zero.

Weak Point Terms
One of the most impressive features of FEMLAB is its ability to specify point
source terms “semi-analytically” through the use of weak terms. Dirac delta
functions, for instance, in the PDE are easily expressed through contributions to
the FEM assembly through simple weak terms. The example in the User’s Guide
of a Poisson equation with a unit point source, or in Chapter 2 of the module
workbook of a point vortex, both show how to implement a point singularity
through weak terms. The only question posed here is what solver parameters are
consistent with the weak source terms.

Solver Parameters

It is probably common sense that the weak solution form should be used on the
Solver Parameters dialogue window if any weak terms or constraints are
implemented. In FEMLAB 2.2, our film drying example worked fine with a
numeric Jacobian and either coefficient, general, or weak solution form. In
FEMLAB 2.3, however, this cocktail produces the error message “Index into
matrix is negative or zero.” The proper solution is found with the “exact
Jacobian” option and weak solution form is recommended. A plausible
explanation for the difference due to the change of versions has not been
forthcoming, so this note should just serve as an advisory that the exact Jacobian
and weak solution form are consistent with pointwise weak terms. COMSOL
advise that the version inconsistency is probably due to the new “Automatic
Scaling of Variables” feature.
The film drying example described here serves as a paradigm for treating
pointwise weak terms. The FEMLAB model here is specified with only base
FEMLAB application modes. It should be understood that by specifying a point
source, the FEM implementation will smooth out the Dirac delta function on a
sub-element length scale. Thus grid resolution may change the influence of the
source at least locally. Thus the solution may never be “grid independent” in all
details, but rather the model of the point source is intimately linked to the
element meshing. Some other test than grid independence must then be passed
to validate the model, typically experimental validation or matching behavior in
limiting cases.