A new arXiv preprint reports numerical tests of two diffuse-interface tumor-growth models in two and three dimensions. In a long nonsmooth GLSS run, the reported maximum Bi-CGSTAB count was 251 over the first 10,000 time steps. In the comparison without preconditioning, Bi-CGSTAB failed to converge within the prescribed maximum of 500 iterations.
The study focuses on the machinery of numerical solving. Its central test was whether block preconditioners based on Schur-complement approximations could remain effective as model parameters, time-step sizes and spatial resolution changed. The numerical scope included the smooth HKNZ formulation and both smooth and nonsmooth GLSS formulations.
The HKNZ model held its line.
In the three-dimensional HKNZ two-tumor experiment, MINRES iteration counts stayed nearly constant through the first 20 time steps. That pattern was reported while mesh refinement, proliferation rate and interface thickness changed. The result is a numerical test of solver behavior across those settings, not a biological measurement.
The HKNZ comparison also found a modest iteration-count advantage for block-triangular preconditioning over the block-diagonal version in both two and three dimensions. HKNZ systems used MINRES. Because exact counts were not supplied for this comparison, the size of the difference remains unclear.
Smooth GLSS showed a similarly steady picture, but used Bi-CGSTAB for its nonsymmetric Newton-linearized systems. Newton converged in only two iterations through almost the entire simulation. Bi-CGSTAB counts rose as the time step became smaller, yet stayed in a relatively small range, and the preconditioners remained effective across the tested time-step sizes and three initial conditions.
AMG had the clearer three-dimensional edge.
The clearest head-to-head result came from a three-dimensional smooth GLSS comparison of AMG and ILU. AMG required roughly 30 to 85 Bi-CGSTAB iterations, while ILU required 80 to 220. ILU also varied more across time steps in that test.
That comparison supports a narrower conclusion than a general ranking of solvers. AMG had the lower and less variable reported iteration range in the tested three-dimensional smooth GLSS configuration only. The evidence does not establish that AMG is universally superior to ILU in other settings.
The nonsmooth case was the harder numerical test.
In the nonsmooth GLSS case, the report describes the linearized system as increasingly ill-conditioned as the Moreau-Yosida parameter c approached zero, with the condition number reported as O(c^-1). The tested final regularization level was cmax = 10^-5, and the time step was also 10^-5. Full continuation was used during the first five time steps; from step six onward, only the final regularization level was solved at each time step.
The phase-field results separated the smooth and nonsmooth formulations. The smooth version showed overshoots up to approximately 1.037. The nonsmooth version stayed within approximately 5 x 10^-5 of the obstacle bounds and reached a minimum of approximately -1.000041 at step 20. In the reported tests, that was tighter obstacle enforcement than in the smooth formulation.
Over the first 10,000 time steps, the nonsmooth run averaged approximately 20,173 degrees of freedom and reached a maximum of 21,257. Its maximum Newton count was 16, alongside the maximum Bi-CGSTAB count of 251. These figures describe one reported long simulation, not a statistical distribution across many runs.
A result about solvers, not patients.
Taken together, the results support a practical claim about the tested numerical setups: Schur-complement block preconditioners remained effective across selected changes in parameters, time steps and spatial resolution, in two and three dimensions. Both formulations used linear finite elements on triangulated meshes and DOLFINx, while spatial adaptivity was applied to the Garcke formulation.
The paper therefore offers evidence about computational convergence and phase-field behavior under selected conditions, rather than evidence about real human, animal or clinical tumor outcomes. Its iteration counts and degrees of freedom were reported for chosen geometries, meshes, parameter settings and simulation horizons, so broader testing would be needed to show whether the same pattern generalizes. The document is an arXiv preprint, version 1, dated 26 August 2026.
Paper data and sources
Original title: Preconditioning for Diffuse Interface Tumor Growth Models
Authors: Jessica Bosch, Pelin Çiloğlu, Chen Greif, Martin Stoll
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text