Non-Matching Mixed-Dimensional Grids

Updated 16 December 2025

Non-matching mixed-dimensional grids are discretization schemes that independently mesh different geometric domains and couple them at interfaces of varying dimensions.
They utilize transfer grids, stable discrete projection operators, and spline-based continuity constraints to enforce coupling and control quadrature errors.
These methods enable robust simulations of multiphysics problems such as fluid-structure interaction, fractured porous media, and linear elasticity while supporting adaptive error estimation.

Non-matching mixed-dimensional grids arise in the numerical approximation of PDEs and geometric modeling where computational domains of different dimensions are coupled and discretized independently, with the resulting meshes or patch structures allowed to be non-conforming at the interfaces. This scenario is prevalent in applications such as fluid-structure interaction, fractured porous media, heterogeneous materials, and multivariate spline theory. The primary challenges are the formulation of variational problems, stable coupling of variables defined on adjacent non-matching subdomains of varying dimensions, assembly and analysis of coupling operators, continuity prescriptions for splines, and the design of computational methods that retain accuracy, stability, and a posteriori error control.

1. Problem Setting and Mathematical Formulations

Mixed-dimensional problems typically involve a partition of a geometric domain in $\mathbb{R}^n$ into subdomains $\Omega_i$ of various intrinsic dimensions $d_i$ (e.g., $d_i\in\{0,1,2,3\}$ for point, curve, surface, and volume), and interfaces $\Gamma_j$ of lower dimension connecting higher- and lower-dimensional subdomains. The discretization process assigns a simplicial or simplotopic mesh to each domain/interface independently, producing non-matching grids at shared boundaries or embedded interfaces (Varela et al., 9 Dec 2025).

A prototypical example is the simulation of Darcy flow in a fractured porous medium, where the matrix, fractures, and their intersections (e.g., 3D, 2D, 1D) are independently meshed. Similarly, fluid-structure interaction relies on separate meshes for the fluid and solid, which may move relative to each other (Boffi et al., 6 Jun 2024).

Key mathematical formulations rely on:

Fictitious domain or mixed-dimensional variational principles, wherein separate variables (e.g., velocity, pressure, displacement, or multiplier fields) are defined and coupled via Lagrange multipliers or constraint operators (Heltai et al., 2023, Boffi et al., 6 Jun 2024).
Saddle-point formulations in Hilbert spaces, analyzing the well-posedness via coercivity and discrete/continuous inf-sup (Ladyzhenskaya–Babuška–Brezzi) conditions. In mixed-dimensional/asymptotic reduction scenarios, robust stability with respect to inclusion size or mesh size ratios is essential (Heltai et al., 2023).
Spline-based continuity constraints for geometric modeling, where the continuity of functions or their derivatives across non-matching simplotopic cells of mixed dimension is expressed in terms of Bernstein–Bézier coefficients and operator differences (Visser et al., 2016).

2. Discretization and Coupling on Non-matching Grids

The treatment of non-matching grids hinges on the definition and assembly of coupling matrices, transfer operators, and interface conditions between disparate meshes. The general principles are as follows:

Coupling matrix assembly between fluid and solid in FSI: the exact assembly requires computing integrals over the intersection regions of solid mesh cells mapped into the fluid mesh, while approximate quadrature is performed directly on the solid mesh using mapped quadrature points (Boffi et al., 6 Jun 2024).
Transfer grid construction for mixed-dimensional PDEs: to mediate non-matching meshes between adjacent subdomains/interfaces, a transfer grid is defined as the common refinement of the two relevant meshes, ensuring each transfer cell is contained in a unique cell on each side. This enables the definition of stable discrete projection operators for potentials and fluxes (Varela et al., 9 Dec 2025).
Stable discrete projections: Operators such as $\widetilde{\mathcal{P}}$ (for trace and primal variables) and $\widetilde{\mathcal{D}}$ (for dual variables and mass-conservative fluxes) are constructed through prolongation onto transfer grids and stable interpolation or local $L^2$ projections (Varela et al., 9 Dec 2025).
Spline continuity enforcement across non-matching simplotopic cells involves embedding each cell in a higher-dimensional circumscribed simplex, stating multivariate Bernstein–Bézier $C^r$ continuity conditions in the ambient simplex, and pulling them back with degree-raise/degree-reduce operators to the original simplotopic setting (Visser et al., 2016).

3. Analysis: Quadrature and Discretization Error, Stability, and Continuity

The substitution of exact coupling integrals with their discrete counterparts, and the independent construction of meshes, generate nontrivial error and stability issues:

Quadrature error in coupling integrals can be bounded via localized Bramble–Hilbert-type estimates. For $L^2$ -type couplings, the error decays as $O(h_S^{3/2} |\log h_S|)$ , and for $H^1$ -type (with derivatives), as $O(h_S^{1/2} |\log h_S| + h_S/h_F)$ , where $h_S$ , $h_F$ are the mesh sizes for solid and fluid respectively. Stability and convergence are retained if $h_S \to 0$ (for $L^2$ ), and both $h_S \to 0$ and $h_S/h_F\to 0$ (for $H^1$ ) (Boffi et al., 6 Jun 2024).
Discrete inf-sup stability for reduced Lagrange multiplier methods in mixed-dimensional coupling is shown to depend delicately on mesh alignment, polynomial degree, and the number of "modes" in the multiplier space. Provided that the ratio $h/\varepsilon$ is sufficiently small, or the number of modes is increased appropriately, stability constants remain bounded (Heltai et al., 2023).
$C^r$ continuity criteria for mixed-dimensional simplotopic grids depend on "out-of-facet parallelism" between adjacent cells and are reduced to algebraic conditions on Bernstein–Bézier coefficients computed via De Casteljau differences and degree adaptation in the out-of-facet factors (Visser et al., 2016).
A posteriori error estimation frameworks for mixed-dimensional Darcy flow employ local cellwise indicators (diffusive/residual) accumulated over different dimensions, and demonstrate fully guaranteed and efficient error bounds under arbitrary (possibly highly non-matching) mesh refinement patterns (Varela et al., 9 Dec 2025).

Discretization method	Main error/stability estimate	Essential mesh criteria
Fictitious-domain FSI (Boffi et al., 6 Jun 2024)	Coupling error $O(h_S^{3/2}\|\log h_S\|)$ or $O(h_S/h_F)$	$h_S/h_F \ll 1$ (for $H^1$ coupling)
Mixed-dimensional PDE (Varela et al., 9 Dec 2025)	Majorant $\mathcal{M}^{\oplus}$ bounds true error	Transfer grid resolves interface
Lagrange multiplier (Heltai et al., 2023)	Inf-sup $\beta_h^{n,\varepsilon} = O((1+h/\varepsilon)^{-n})$	$h/\varepsilon$ small or large $n$

4. Numerical Techniques and Implementation Algorithms

Computational strategies rely on explicit steps to ensure both accuracy and practical feasibility:

Quadrature strategy: For non-matching coupling in FSI, a degree-2-exact quadrature (e.g., 3–4 points for triangles) on the solid mesh suffices for $L^2$ -type weak coupling, while $H^1$ -type coupling demands either finer solid meshes or higher-order quadrature rules, and $h_S/h_F \leq 0.5$ (Boffi et al., 6 Jun 2024).
Transfer grid assembly: Non-empty intersections of interface and adjacent subdomain mesh elements are detected, subdivided into simplices as needed, and collected to form the transfer grid used for projections (Varela et al., 9 Dec 2025).
Spline constraint assembly: For each adjacent cell pair, the facet and out-of-facet factors are detected, De Casteljau differences and degree adjustments are computed, and constraints on Bernstein–Bézier coefficients are generated for enforcement of $C^r$ across cell interfaces (Visser et al., 2016).
A posteriori error assembly: Local error indicators are computed for each mesh cell and interface segment using available numerical solution and projections, yielding global majorant and lower local indicators suitable for mesh refinement guidance (Varela et al., 9 Dec 2025).

Example pseudocode for assembling $C^r$ constraints in mixed grids (Visser et al., 2016):

for each adjacent cell pair (cell_A, cell_B):
    (i, j) = find_out_of_facet_factors(cell_A.type, cell_B.type)
    sA = unit_vector_in_factor(cell_A.simplex[i])
    sB = unit_vector_in_factor(cell_B.simplex[j])
    for multi-index kappa with sum(kappa) = d - r and kappa[i, end] = r:
        left = de_casteljau(cell_A.BB, kappa, sA, r, factor=i)
        left_adj = degree_raise(left, r, factor=j)
        kappa_prime = reindex_for_B(kappa, i->j)
        right = de_casteljau(cell_B.BB, kappa_prime, sB, r, factor=j)
        right_adj = degree_reduce(right, r, factor=i)
        add_constraint(left_adj, right_adj)

5. Practical Guidelines and Benchmark Results

Empirical and theoretical studies provide the following prescriptions:

Mesh refinement: Keep the mesh size in lower-dimensional models (e.g., fractures or inclusions) comparable to, or finer than, adjacent higher-dimensional meshes to preserve stability and avoid large projection errors (Boffi et al., 6 Jun 2024, Varela et al., 9 Dec 2025).
Quadrature rule selection: Use degree-2-exact rules for $L^2$ couplings; higher for $H^1$ or derivative couplings; avoid excessive $h_S/h_F$ ratios (Boffi et al., 6 Jun 2024).
Projection and transfer grid design: Always maintain shape regularity of meshes, build transfer grids that resolve intersection geometry accurately, and refine meshes on both sides of interfaces congruently to preserve inf-sup stability (Varela et al., 9 Dec 2025).
Mode selection in model reduction: For highly slender inclusions, a small number of multiplier modes suffices; for $h \approx \varepsilon$ , more modes are needed. For local features or pointlike inclusions, empirical accuracy is already high with $N=1$ or $N=3$ (Heltai et al., 2023).
Error estimator interpretation: Local indicators identify dominant regions of error (bulk vs. interface); in experiments, global majorants closely track the true error even in highly non-matching grid scenarios, guiding adaptive refinement (Varela et al., 9 Dec 2025).

6. Extensions and Generalizations

The frameworks surveyed adapt robustly to a broad class of mixed-dimensional and multiphysics problems. The key ingredients—elliptic bilinear forms, H(div)–L $^2$ and trace couplings, local mass conservation, and transfer/projection operators—support extension to:

Linear elasticity with discrete inclusions (e.g., fiber-reinforced or truss/plate structures),
Reaction-diffusion in layered or stratified domains,
Electromagnetics with conducting sheets or wires,
Thermo-poroelasticity in fractured media (Varela et al., 9 Dec 2025).

A plausible implication is that the majorant-based a posteriori error estimates, transfer grid techniques, and out-of-facet parallelism continuity analysis will provide foundational tools for future developments in non-matching mixed-dimensional discretizations across multiple fields.

Key References: (Varela et al., 9 Dec 2025, Boffi et al., 6 Jun 2024, Heltai et al., 2023, Visser et al., 2016)