Intro to surfacefun#

An interactive introduction to surfacefun — a package for numerically computing with functions on surfaces, and solving PDEs on them, with high-order accuracy. Everything here runs in your browser through numbl: pick a script on the left and click Run. Each script is self-contained and draws one or more figures (use the figure tabs).

Every script begins with

mip load --install flatironinstitute/flatironinstitute/surfacefun

which fetches surfacefun (and its dependency, Chebfun) on first use.

1. Surface meshes#

A surfacemesh is a collection of high-order quadrilateral patches whose union defines a surface in 3-D.

• mesh_single_patch.m — build a surface from one high-order Chebyshev patch.
• mesh_sphere.m — the built-in "cubed sphere", plus mesh queries (patches, order, area, volume, bounding box).
• mesh_blob.m — a smooth random "blob".
• mesh_torus.m — a genus-1 torus.
• mesh_twisted_torus.m — patches need not join smoothly.
• mesh_mobius.m — surfaces need not be orientable.
• mesh_refine.m — change the order (resample) or split patches (refine).

2. Scalar functions#

A surfacefun is a scalar field on a surfacemesh, built from a handle of the Cartesian coordinates (x,y,z).

• fun_plot.m — define and plot a function on the sphere.
• fun_arithmetic.m — overloaded +, *, abs, ...
• fun_contour.m — contour (level-set) plot on the surface.
• fun_gradient.m — the surface gradient grad(f).
• fun_laplacian.m — the surface Laplacian lap(f).
• fun_integral_norm.m — integral, mean, and norms.

3. Vector fields#

A surfacefunv is a three-component vector field over a surfacemesh.

• vec_normals.m — the unit normal field, as a quiver plot.
• vec_gradient.m — grad(f) is tangent to the surface.
• vec_mean_curvature.m — mean curvature from div(n)/2.
• vec_curl.m — the surface curl of a vector field.

4. Solving PDEs#

surfaceop is a fast direct solver for second-order linear elliptic PDEs on surfaces.

• pde_laplace_beltrami.m — a Laplace-Beltrami problem on the sphere, checked against an exact spherical-harmonic solution.
• pde_helmholtz.m — a variable-coefficient Helmholtz problem on a stellarator.
• pde_open_surface.m — an open surface with Dirichlet boundary data, reusing one factorization for several right-hand sides.

5. Worked examples#

• ex_hodge.m — the Hodge decomposition of a tangential vector field into curl-free, divergence-free, and harmonic parts.
• ex_reaction_diffusion.m — the complex Ginzburg-Landau equation, integrated in time with a reused implicit solver.

Notes#

The scripts keep the polynomial order and refinement modest so they run quickly in the browser. Raise p and nref for higher accuracy and smoother pictures. The numerics follow the surfacefun documentation at https://surfacefun.readthedocs.io.

How it's deployed#

On every push to main, the workflow in .github/workflows/deploy.yml bundles these files with the numbl browser IDE and publishes the result to GitHub Pages. Edit numbl-project.json to change the title or which file opens first.