Error Estimation
The Error Estimation node () appears under the solver node when you use goal-oriented error estimation (see Error Estimation — Theory and Variables) and when Adaptation and error estimates in the defining study step is not set to None.
General
From the Defined by study step list, choose any applicable study step or choose None. If you choose None, no settings are displayed here, and none of the otherwise available setting values are used.
The Error estimate list is set in the parent study step list and affects the settings available here:
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If it is L2 norm of error squared, use the Scaling factor field to enter a space-separated list of scaling factors, one for each field variable (default: 1). The error estimate for each field variable is divided by this factor. Also, the L2 norm error estimate is based on a stability estimate for the PDE. Use the Stability estimate derivative order field to specify its order (default: 2). For certain problems, which are symmetric and where strong error estimates hold, this method is equivalent to a functional error estimation with the functional being the L2 norm squared of the solution. This method can be used also for problems where these assumptions do not hold, but then the adaptation will not be optimal. See the following for some more in-depth information about these settings.
The L2 norm of error squared method estimates the error for a mesh element as a summation of contributions for the different equations solved for. It sums over
where A is the element area (volume, length), h is the element size, q is the Stability estimate derivative order, s is the Scaling factor, and ρ is an estimate of the PDE residual. The asymptotic behavior for ρ is that it is proportional to hp, where p is the Residual order (see below). Even if it is possible to estimate the actual value for ρ very generally, assisting the algorithm with this order is important — for example, for the Element selection method Rough global minimum (in the Adaptive Mesh Refinement subnode; see Adaptive Mesh Refinement (Stationary and Eigenvalue Adaptation)), which essentially solves an optimization problem for where to refine so that the total error is reduced as much as possible (constrained by how many elements that can be added). All of the values for the Stability estimate derivative order, Scaling factor, and Residual order (q, s, and p, respectively) can be given as vectors for the different equations. For the Stability estimate derivative order, the default is 2, and it is related to the stability estimate that holds for the problem at hand. If it is not of a Poisson type, then it might need adjustment. For the Scaling factor, the default is 1. It is mainly important to relatively weigh in the different parts of the equations solved for, which means that you need to give a space-separated array of numbers in general. Notice that when solving a multiphysics problem, the summation over different equations will not have the same unit since the different ρ will have different units. So, a first effort with scaling would be to take this into account. In fact, not even a single-physics case like fluid flow will make this summation unit consistent because the residual for momentum and mass conservation will be added up with the default scaling factors of 1. For the Residual order, the default is one order lower than the shape functions used for the equation. Change this only for nonstandard PDEs. Roughly speaking, the expected order is the basis order minus the highest spatial derivative order in the weak formulation. For a second-order PDE formulation with an integration by parts (lowering all second-order derivatives to first order) this will be of order one.
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If it is Functional and specify a Functional type. Available functional types are Predefined and Manual. This option adapts the mesh toward improved accuracy in the expression for the functional (for example, some energy, drag, or lift). Select Manual to specify a globally available scalar-valued expression in the Functional field (for example, the name of a global variable probe). If you select Predefined, you can choose from a predefined list of functional from the Solution functional list:
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 Integral (the default)
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 L2 norm
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 L1 norm
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 Approximate max norm
For a Stationary study step stat, a global variable is defined for the functional with the name stat.gfunc (and similarly for a frequency-domain study step). The functional can be evaluated under Results>Derived Values by adding a Global Evaluation node and, in the Expressions section in its Settings window, selecting Global Definitions>
Error estimation>stat.gfunc - Functional - Stationary.
The functional must be differentiable (or complex-valued analytic). Also, the expressions in the formulation must be differentiable. If this does not hold, the adjoint solution and its error estimate risk not being accurate, and the adaptation will then not be optimal for the functional used. The Functional option estimates the error for a mesh element as a summation of contributions. It sums over A ωK ρ, where ωK is computed from the adjoint (or dual) solution. ωK can be computed with different methods (PPR for Lagrange or Interpolation error). Equation 20-5 is about how ωK is computed for PPR for Lagrange. Notice that this method is not using scaling factors or the stability estimate derivative order because they are not part of the formula (they are built into ωK). But ρ is part of the formula and the residual order can therefore be important (for example, for the Rough global minimum method; see below). For the Approximate max norm, high order p-norm. The reason for this is that it is possible to differentiate this functional.
By default, the software automatically determines the order of decrease in equation residuals on basis of the shape function orders in the geometry. To specify a residual order manually, select the Residual order check box and specify a nonnegative integer in the accompanying field (available when Error estimate is L2 norm of error squared or Functional).
From the Adjoint solution error estimate list (only available when Error estimate is Functional), select an error estimate method for the adjoint solution. Select PPR for Lagrange (the default) to use the polynomial preserving recovery technique when possible (that is, for fields represented using Lagrange elements). Select Interpolation error to always use an estimate based on the interpolation error. For details, see The Functional Error Estimate.
Advanced
From the Compensate for nojac terms list, choose Automatic (the default), On, or Off.
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With Automatic, the software tries to assemble the complete Jacobian if an incomplete Jacobian has been detected. If the assembly of the complete Jacobian fails or in the case of nonconvergence, a warning is written and the incomplete Jacobian is used in the sensitivity analysis for stationary problems. For time-dependent problems, an error is returned.
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With On, the software tries to assemble the complete Jacobian if an incomplete Jacobian has been detected. If the assembly of the complete Jacobian fails or in the case of nonconvergence, an error is returned.
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If you get a warning about an incomplete Jacobian, you can then avoid that warning choosing Off from this list. With that setting, the software does not attempt to assemble the complete Jacobian (the incomplete Jacobian is used immediately).