General-form PDE

Node.js examples of generalFormPDE, in examples/generalFormPDE. Run the commands below from that folder (see running an example). All results are from FEAScript-core 0.4.0.

Advection-diffusion in 1D

Steady advection-diffusion on $[0, 1]$ with a Gaussian source centered at $x = 0.5$:

$$\frac{d^2u}{dx^2} - 10 \, \frac{du}{dx} = 10 \, e^{-200 (x - 0.5)^2},$$

that is $A = 1$, $B = -10$, $C = 0$ and $D(x) = 10 \, e^{-200 (x - 0.5)^2}$ in the general form. The value is fixed at the inlet, $u(0) = 1$, and the gradient is zero at the outlet.

node advectionDiffusion1D/advectionDiffusion1D.js

Source: advectionDiffusion1D.js

Geometry

Mesh

Results

Upstream of the source the solution stays close to the inlet value. Across the source it drops, and downstream it is carried unchanged to the outlet, where $u(1) = 0.8756$. The markers are the end nodes of the elements. The example prints the node coordinates and the solution vector.

Nonlinear reaction-diffusion in 1D

Steady diffusion with a quadratic reaction on $[0, 10]$:

$$\frac{d^2u}{dx^2} - u^2 = 0,$$

solved with the Newton-Raphson method (nonlinear: true, with $D(x, u) = u^2$ and its derivative $\partial D / \partial u = 2u$). The value is fixed at $u(0) = 1$ and the gradient is zero at $x = 10$.

node nonlinearReactionDiffusion1D/nonlinearReactionDiffusion1D.js

Source: nonlinearReactionDiffusion1D.js

Geometry

Mesh

Results

The reaction consumes $u$ as it diffuses into the domain: on the 20 linear elements of the example, $u = 0.5009$ at $x = 1$ and $u = 0.0568$ at $x = 10$. The elements are 0.5 long, so these values are 0.7 % and 0.5 % below the mesh-converged solution, $u(1) = 0.5043$ and $u(10) = 0.0571$, which FEAScript reaches on finer meshes. The example prints the node coordinates and the solution vector.