Scalar Kernels

  • Scalar Kernels:
    • Operate on scalar variables (family = SCALAR).
    • Are defined in the [ScalarKernels] section of your input file.
  • Use them for:
    • Solving ODEs (see example 18).
    • Formulations with Lagrange multipliers.
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    • Other applications...
  • Notes:
    • Mesh-specific data such as qp and currentelem are not available to ScalarKernels.

  • Problem being solved:

$$$$\begin{aligned} \frac{\partial u}{\partial t} &= \nabla^2 u + f && \text{in }\Omega = [-1, 1] \\ u &= X(t) && \text{on }\Gamma_{\text{left}}\\ u &= Y(t) && \text{on }\Gamma_{\text{right}}\\ \end{aligned}$$$$

  • Where the boundary conditions are governed by the ODEs:

$$$$\begin{aligned} \frac{dX}{dt} &= 3X + 2Y \\[6pt] \frac{dY}{dt} &= 4X + Y \\ \end{aligned}$$$$ plus suitable initial conditions.

Example 18