Learn PyMHM by method

Start with the feature and API overview, then choose a method. Every lesson follows meshes → local problem → global problem → assembly → solve → fields and errors, with the mathematical formulation next to its code. Ready-made physical solvers appear only as optional conveniences after the definition.

A final convergence study identifies the physical observable, refinement variable, approximation spaces and hypotheses of its cited estimate. A polynomial patch checks the implementation; the refinement studies use solutions outside the chosen local spaces and several terminal intervals to examine the asymptotic regime.

Convergence plots

Each refinement figure preserves the coarse levels. Dashed curves show the cited target powers anchored to the finest measured error. The successive-rate panel uses the actual refinement ratios, and a third panel shows the error amplitude divided by the target power:

\[ \begin{aligned} r_i&=\frac{\log(E_{i-1}/E_i)}{\log(h_{i-1}/h_i)},\\ A_i&=\frac{E_i}{h_i^q}. \end{aligned} \]

Stable consecutive rates near \(q\), together with a plateau in \(A_i\), distinguish the asymptotic regime from a single favorable slope. The shaded window always contains the last four measured levels. The accompanying table reports its three successive orders and the variation of its normalized amplitude. Each page identifies whether its target comes from an error estimate or a published numerical observation. Spatial studies refine the macro or trace scale under the stated hypotheses; time studies keep spatial resolution fixed. The convergence notebook recomputes these diagnostics from the stored records.

Local and global formulations

Method	Unknowns
Primal MHM	pressure and normal Darcy flux
MHM with mixed H(div) locals	Darcy flux, pressure and boundary pressure
Robin multiscale hybrid method	pressure and Robin multiplier
MH²M: pressure, conormal and trace	pressure, private conormal and continuous pressure trace
MsHHO: cell and face moments	cell/face pressure moments and energy reconstruction
Petrov–Galerkin MHM	base and enriched pressure with normal flux
MHM-USFEM	scalar values, coarse moments and normal multiplier
Unfitted MHM traces	pressure and independently segmented normal-flux traces
Stokes–Brinkman MHM	velocity, pressure and pseudo-traction
Oseen MHM	velocity, pressure and half-advection pseudo-traction
Primal elasticity MHM	displacement and physical traction
Displacement–pressure GaLS MHM	displacement, pressure and negative physical Cauchy traction
Mixed H(div) stress MHM	stress rows, displacement and weak rotation
Transient transport MHM	time-dependent scalar and interface multiplier
Elastodynamic MHM	displacement, velocity, acceleration and traction
Helmholtz MHM	complex pressure and normal multiplier
Maxwell MHM	electric, magnetic and tangential trace fields

Reconstruction, indicators and refinement

Strategy	What you will verify
Flux and potential reconstruction	recovered-field convergence, normal continuity, projected divergence and continuous-test equilibrium
Error indicators	physical residuals, estimator effectivity and recovery contributions
Adaptive spaces	error decrease against work with independent scale controls
Recursive MHM	equivalent nested condensation, inherited boundary and gauge

Explore physical problems in the Gallery or select an execution environment. All cited publications are collected in the Bibliography.