A 64³ parametric diffusion problem. Each slider move takes tens of milliseconds on a CPU; the full solve takes about a second. Drag the cubes to rotate them; the three slice sliders move the cutting planes. All demos
$$-\nabla\!\cdot\big(e^{a(y,x)}\,\nabla u\big) = 1 \ \text{ in } (0,1)^3,\qquad u = 0 \ \text{ on the boundary},\qquad a(y,x) = \sum_{j=1}^{200} y_j\, j^{-s}\, \psi_j(x),\quad \psi_j(x) = \prod_{i=1}^{3}\sin(k_{j,i}\pi x_i).$$
Parameters: \(y \in [-1,1]^{200}\) (sliders \(y_1,\dots,y_6\); the rest 0 or random), and \(s\) when the slider is shown. Grid: \(64^3\) cells.