An adapted solution at t = tn is mapped to the coarse base mesh. A new adapted mesh for the time interval
[tn, tn+1] is constructed by first computing a coarse solution on the base mesh in
[tn, ts], where
ts is the largest sample time and
tn+1 = 2tn − tn−1. The error indicator is evaluated using the coarse solution at the given sample points.
The new adapted mesh is obtained by using the error indicator sampled at given points in [tn, tn+1], selecting a set of elements based on the element pick function, and then finally refining these elements. The solution at
tn to the PDE problem on the previous adapted mesh for
[tn−1, tn] is then mapped to the new mesh for
[tn, tn+1] and time integration continues until the next mesh adaptation takes place at
tn+1.
Here γ is an
N-vector of integers containing the number of times the element at that position should be refined,
p = max
iγ(i), and
N is the number of elements of the coarse base mesh.
Now, the next interval length is decreased by a given factor if ρ is larger than 120% of the requested reference value. If
ρ is smaller than 80% of the reference value it is instead increased. Otherwise, the interval length is kept the same.
The L2 norm error estimate relies on an assumption of a strong stability estimate for the PDE problem (satisfied, for example, for Poisson’s equation over a domain with a smooth boundary). From such an assumption, it is possible to show that there is a constant
C, such that the
L2 norm of the error,
el, in the
lth equation satisfies
where ρl is the residual in the
lth equation and
ql is the stability estimate derivative order.
h is the local mesh element size. The adaptive solver algorithm uses the following
L2-norm error indicator:
with the default value ql = 2. This formula also introduces the scaling factors
sl for the residual with the default value
sl = 1. The local error indicator for a mesh element is
where A is the area (volume, length) of the mesh element, and
τl is the absolute value of the
lth equation residual (one number per mesh element).
where el* and
ρl are the error in the dual or adjoint solution to, and the residual for, the
lth equation, respectively. The adaptive solver algorithm uses the following error indicator for a mesh element:
where A is the area (volume, length) of the mesh element, and
τl is the absolute value of the
lth equation residual (one number per mesh element). Here
wl is an estimate of the adjoint solution error for the
lth equation. This error is estimated using the following methods. For all three methods, the sensitivity solver is used to find the discrete adjoint solution.
A possible error indicator is the L2 norm of the gradient of the dependent variables (for example,
sqrt(comp1.Tx^2+comp1.Ty^2) for the temperature in a 2D heat transfer model). The gradient of the dependent variable is the default value for the error indicator in most physics interfaces.
A solution on the coarse base mesh is computed in the next time interval, and the error indicator is evaluated at the points specified in the Sample points field. In this way a new adapted mesh appropriate for the next time interval can be generated. The sample points must be specified as a number between 0 and 1 because they are interpreted as being relative to the time interval under consideration. Entering a scalar value of 0.5 means that the error indicator is evaluated at the midpoint of the interval. The default value is
range(0.0,0.1,1.0), which gives 11 sample points from 0 to 1.