Galerkin tools

UnifiedSparseGrids provides a small set of building blocks for matrix-free sparse grid Galerkin workflows. They are designed to work with coefficients stored in the package's recursive layout and to compose efficiently with the Unidirectional principle.

Matrix-free tensor-sum operators

Many PDE operators can be written as a sum of separable tensor terms, for example

\[(-\Delta + \alpha) = \sum_{j=1}^D I \otimes \cdots \otimes K_j \otimes \cdots \otimes I \; + \; \alpha\,(I \otimes \cdots \otimes I),\]

where each K_j is a 1D stiffness-like operator in dimension j.

The generic helper types are:

• WeightedTensorTerm(w, op) for a weighted AbstractTensorOp term,
• TensorSumMatVec(grid, terms, T) for a mul!-compatible matrix-free matvec applying a sum of terms,
• as_linear_operator(A) to wrap a TensorSumMatVec as a LinearOperators.LinearOperator for Krylov solvers.

These are used throughout the examples to define operators without forming large dense matrices.

Hat-basis line operators (Balder–Zenger)

For the hierarchical hat basis used in the Balder–Zenger Helmholtz example, the package provides specialized 1D building blocks:

• HatMassLineOp (with updown(op) support for triangular splitting),
• HatStiffnessDiagLineOp.

The key design point is that these line operators are matrix-free and in-place: they implement apply_line! for a single fiber and are composed into a tensor operator via TensorOp/ UpDownTensorOp.

Discrete error utilities

Examples often validate solutions by comparing discrete samples against a reference function uex. This is supported by discrete_l2_error, which is available at three convenience levels:

• High-level: discrete_l2_error(u, grid, basis, uex, ref_level; eval_nodes=...). Builds a tensor-product reference point set and evaluates the sparse grid expansion.
• Medium-level: discrete_l2_error(u, grid, basis, uex, eval_points). Evaluates via the evaluation interface (plan_evaluate + evaluate).
• Low-level: discrete_l2_error(uvals, uex, eval_points). Here uvals is interpreted as already aligned with eval_points.

All variants return (L2, relL2) where L2 is the root-mean-square error and relL2 is the relative RMS error.