Remarks on Solver-Related Model Characteristics

The Importance of a Correct Jacobian Matrix

The solvers break down each problem — linear or nonlinear — into one or several linear systems of equations by approximating the given problem with a linearized problem. The coefficient matrix of the discretized linearized problem is called the Jacobian matrix (or stiffness matrix). In most cases COMSOL Multiphysics computes a correct Jacobian matrix.

The consequences of an incorrect Jacobian matrix depend on the solver in use:

•	The linear stationary solver and the eigenvalue solver simply give an incorrect solution.

•	The nonlinear stationary solver and the time-dependent solver take longer time to converge to the correct solution, and in some cases the solver even fails to find a solution. However, if the ignored terms have a very weak dependence on the sought solution, the impact on convergence speed is small.

An incorrect Jacobian matrix can occur in the following cases:

•	If you supply an incorrect derivative of some of the user-defined functions and then use that function in some PDE coefficient or boundary condition specification with arguments that depend on the solution (for example, with the temperature as the function argument in a heat-transfer simulation).

•	If you use a MATLAB M-file function or the External function interface, for which you have specified no derivative, and then call this function in some PDE or boundary condition with arguments that depend on the solution. A correct Jacobian can be computed if the function derivative is defined.

•	If you use nonanalytic functions in a complex-valued problem, such as real, imag, conj, abs, or arg.

•	If you use the nojac operator.

Working with Complex-Valued Problems

When a problem contains complex numbers, be sure to consider the following two aspects:

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For time-dependent problems, the time-stepping algorithm must know if a problem is complex valued. If your model uses a complex-valued initial solution, the COMSOL software detects this and classifies the problem accordingly. If the problem is complex-valued because of a complex-valued PDE coefficient or other material property, go to the Time-Dependent Solver node’s Settings window, and in the Advanced section select the Allow complex numbers check box.

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If you expect to receive complex outputs from real inputs in elementary functions such as sqrt, log, and asin, or when evaluating a nonintegral power of a negative number, add an Advanced subnode. Then in the Settings window under General, select the Allow complex-valued output from functions with real input check box. By default, the COMSOL software gives an error message if a real input to a function generates a complex output.