The COMSOL Equation Module is a simulation application built entirely in the COMSOL Multiphysics(R) 6.4 Application Builder (pure Java). It turns the expert-only Weak Form PDE interface into a tool any base-license user can drive, extending a basic COMSOL license to custom partial differential equations.

How it works: The user types a PDE in ordinary textbook notation, for example -div(k*grad(u)) = f, curl(curl(A)) = lambda*A, the constrained system curl(H) = J with div(H) = 0, or laplacian(u) + 5*i*u = lambda*u. A custom Java parser tokenizes and classifies the equation (source, eigenvalue, coupled, scalar or vector), then automatically derives the corresponding weak form: test functions, integration by parts, and the correct finite-element basis, Nedelec edge elements for curl operators, Lagrange for grad/div. It builds the geometry, lets the user assign boundary conditions by clicking faces interactively across nine built-in geometries (rectangle through sphere), runs the appropriate stationary or eigenvalue study in 2D or 3D, and returns validated, physically meaningful results.

What makes it novel: Solving a custom PDE in COMSOL normally demands deep weak-form expertise, hand-derived test functions, element-type choices, gauge or penalty terms, and careful handling of sign conventions and spurious modes. The Equation Module automates that entire translation. Its decisive result is in 3D vector eigenvalue problems: using Nedelec edge elements, the enormous null space of the curl-curl operator collapses to a clean machine-zero cluster, isolating the physical spectrum without the spurious modes that corrupt nodal discretizations, behavior normally reserved for dedicated RF or AC/DC modules. The same equation box serves source, eigenvalue, constrained-vector, and complex-spectrum problems, distinguished automatically by structural signature.

How it would be produced: As pure software with no external dependencies, the module is already built and validated through 53 tracked development versions. Production is regression testing against exact analytical benchmarks: the scalar Helmholtz ladder in 2D and 3D; 2D magnetostatics, curl(H) = J under the div(H) = 0 gauge; the curl-curl PEC cavity spectrum of a unit cube, matched to within 0.001% with correct mode ordering and degeneracy multiplicities; a convergence study through the 8*pi^2 band with errors below 0.005%, far inside the 0.1% target; and geometry-independence confirmed on cube, cylinder and sphere. For distribution it deploys unchanged through COMSOL Server(R) or compiles to a standalone executable with COMSOL Compiler(TM), making it instantly scalable worldwide.

Where it would be applied: The market is researchers, educators, and engineers who must model custom or multiphysics phenomena but lack, or cannot justify, specialized modules. By accepting equations in natural notation and removing the weak-form translation step, the module cuts setup time and eliminates a major class of modeling errors. It works equally as a teaching tool for finite-element and PDE courses and as a rapid-prototyping environment in the R&D laboratory. It complements rather than replaces COMSOL's specialized modules: for production-grade AC/DC and RF work, with their full material libraries, multiphysics couplings, and tuned solvers, those modules remain essential, and the Equation Module serves as a natural gateway to them.