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arxiv: 2509.25115 · v2 · submitted 2025-09-29 · 🧮 math.NA · cs.NA

Diffuse Domain Methods with Dirichlet Boundary Conditions

Luke Benfield , Andreas Dedner This is my paper

Pith reviewed 2026-05-18 11:54 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords diffuse domain methodDirichlet boundary conditionsNitsche methodphase-field functionmixed formulationcoercivityNavier-Stokes
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The pith

New diffuse domain methods reformulate Dirichlet boundary conditions as natural conditions for PDEs on complex domains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops alternative ways to solve partial differential equations on intricate geometries without creating custom meshes. Instead of fitting a grid to the domain, it uses a smooth phase-field function on a larger simple domain to represent the boundary. Two new methods are derived from the mixed formulation of the equations, which changes the essential Dirichlet conditions into natural boundary conditions. Coercive formulations based on Nitsche's method are also created, with proofs of coercivity provided for the new and existing methods. Numerical tests show better accuracy and demonstrate use on incompressible Navier-Stokes equations.

Core claim

The authors derive two new diffuse domain methods from the mixed formulation of the governing equations. This transforms the essential Dirichlet boundary conditions into natural boundary conditions. They also develop coercive formulations using Nitsche's method and prove coercivity for all approximations considered. Numerical experiments confirm improved accuracy of the new methods and illustrate the trade-off between L2 and H1 errors, with application to the incompressible Navier-Stokes equations on benchmark problems.

What carries the argument

The mixed formulation of the governing equations, which converts essential Dirichlet conditions into natural boundary conditions, combined with Nitsche's method for coercive formulations.

If this is right

  • Improved accuracy is achieved compared to existing diffuse domain approximations.
  • The balance between L2 and H1 errors can be controlled through the choice of method.
  • The approach enables simulation of incompressible Navier-Stokes equations on complex domains without fitted meshes.
  • Coercivity guarantees stability for the new approximations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These methods could extend to other types of boundary conditions or nonlinear problems by similar reformulations.
  • Adaptive choice of the interface width parameter might further optimize the error balance in practical computations.
  • Comparison with traditional mesh-based methods on the same problems would quantify the computational savings from avoiding mesh generation.

Load-bearing premise

The phase-field function accurately represents the complex geometry on the larger domain with approximation errors that stay controlled as the interface width goes to zero.

What would settle it

A numerical experiment on a simple domain where the new methods show no accuracy improvement over standard DDM or where coercivity fails to hold in practice.

Figures

Figures reproduced from arXiv: 2509.25115 by Andreas Dedner, Luke Benfield.

Figure 1
Figure 1. Figure 1: Arc shape domain. These results are shown in figures 2 and 3. NSDDM (Method 5) almost always has the best L 2 errors, 10 4 × 10−3 6 × 10−3 −2 h 10−2 10−1 100 Arc Error 10 4 × 10−3 6 × 10−3 −2 h −0.5 0.0 0.5 1.0 1.5 2.0 Arc EOC Order: 1 Order: 2 L2error H1error NSDDM Mix0DDM [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diffusion Dominated problem error plot for Arc domain. [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Diffusion Dominated problem error plot for Inverted Arc domain. [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Advection Dominated problem error plot for Arc domain. [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Advection Dominated problem error plot for Inverted Arc domain. [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Solution for searchlight with arc domain. [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Advection problem sample for Arc domain. [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Contour plots of Mix0DDM approximation solutions for Arc domain. [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Contour plots of NSDDM approximation solutions for Arc domain. [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Cylinder Flow Domain of the smallest elements is hmin = 0.005. The element size increases linearly, starting at a distance of 5ϵ = 0.0875 from the interface of the cylinder, reaching the largest size at 20ϵ = 0.35. To ensure numerical stability and reduce time step errors, we set τ = 0.005. This gives the results for the chosen measurements in figure 11. To evaluate the accuracy of these results in table … view at source ↗
Figure 11
Figure 11. Figure 11: Measurements for Navier-Stokes flow around cylinder. [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
read the original abstract

The solution of partial differential equations (PDEs) on complex domains often presents a significant computational challenge by requiring the generation of fitted meshes. The Diffuse Domain Method (DDM) is an alternative which reformulates the problem on a larger, simple domain where the complex geometry is represented by a smooth phase-field function. This paper introduces and analyses several new DDM methods for solving problems with Dirichlet boundary conditions. We derive two new methods from the mixed formulation of the governing equations. This approach transforms the essential Dirichlet conditions into natural boundary conditions. Additionally, we develop coercive formulations based on Nitsche's method, and provide proofs of coercivity for all new and key existing approximations. Numerical experiments demonstrate the improved accuracy of the new methods, and reveal the balance between $L^2$ and $H^1$ errors. The practical effectiveness of this approach is demonstrated through the simulation of the incompressible Navier-Stokes equations on a benchmark fluid dynamics problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper develops new Diffuse Domain Methods (DDM) for elliptic and Navier-Stokes problems with Dirichlet boundary conditions on complex domains. It derives two methods from mixed formulations that convert essential Dirichlet conditions into natural boundary conditions, constructs coercive Nitsche-type variants, supplies coercivity proofs for the new and selected existing DDM approximations, and reports numerical experiments showing improved accuracy together with a favorable L²/H¹ error balance on benchmark problems.

Significance. If the coercivity constants remain controllable as the diffuse-interface width ε tends to zero, the work supplies a theoretically grounded route to stable, high-accuracy DDM discretizations that avoid body-fitted meshes. The explicit coercivity proofs and the observed error balance constitute concrete strengths that would strengthen the practical utility of phase-field representations for incompressible flow.

major comments (3)
  1. [§3.2, §4] §3.2 (mixed-formulation method) and §4 (Nitsche coercivity proofs): the coercivity estimates are stated for fixed ε; the dependence of the lower bound on ε is not quantified. If the constant deteriorates as 1/ε or worse, the a-priori estimates used to justify convergence to the sharp-interface Dirichlet problem become invalid even when the phase-field geometry is exact.
  2. [Theorem 4.3] Theorem 4.3 (coercivity of the new Nitsche-DDM form): the proof invokes a trace inequality whose constant depends on the gradient of the phase field φ_ε. No explicit bound is given showing that this constant remains O(1) or O(ε^α) with α ≥ 0 as ε → 0, which is required for the method to be uniformly stable in the diffuse-interface limit.
  3. [§5.3] §5.3 (Navier-Stokes numerical experiments): the relation between mesh size h, time step, and interface width ε is not stated. Without this relation it is impossible to assess whether the reported L²/H¹ error balance persists under the refinement path needed for convergence to the sharp geometry.
minor comments (2)
  1. [§2] Notation for the phase-field function φ_ε is introduced without an explicit statement of the scaling of its transition layer width with ε.
  2. [Figure 3] Figure 3 caption does not indicate the precise value of ε used for the plotted solutions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for the thorough review and valuable suggestions. Below we provide point-by-point responses to the major comments. We plan to incorporate several clarifications and additional analyses in the revised manuscript.

read point-by-point responses

  1. Referee: [§3.2, §4] §3.2 (mixed-formulation method) and §4 (Nitsche coercivity proofs): the coercivity estimates are stated for fixed ε; the dependence of the lower bound on ε is not quantified. If the constant deteriorates as 1/ε or worse, the a-priori estimates used to justify convergence to the sharp-interface Dirichlet problem become invalid even when the phase-field geometry is exact.

    Authors: We agree that the ε-dependence of the coercivity constants merits further clarification to support convergence analysis as ε tends to zero. The proofs in the manuscript establish coercivity for each fixed positive ε, with the constants potentially depending on ε through the phase-field properties. In the revised version, we will add a new subsection or remarks quantifying this dependence, demonstrating that the constants remain of order O(ε^{-1}) at worst, which is still compatible with standard a-priori estimates for diffuse domain approximations. This will address the concern regarding the validity of the estimates in the sharp-interface limit. revision: yes

  2. Referee: [Theorem 4.3] Theorem 4.3 (coercivity of the new Nitsche-DDM form): the proof invokes a trace inequality whose constant depends on the gradient of the phase field φ_ε. No explicit bound is given showing that this constant remains O(1) or O(ε^α) with α ≥ 0 as ε → 0, which is required for the method to be uniformly stable in the diffuse-interface limit.

    Authors: Thank you for highlighting this aspect of the proof. The trace inequality employed in Theorem 4.3 indeed involves a constant influenced by ||∇φ_ε||_∞, which scales as O(1/ε) within the diffuse interface. We will revise the proof to include an explicit bound, showing that the resulting coercivity constant for the Nitsche-DDM form is O(ε^{-1/2}) as ε → 0. This bound ensures uniform stability in the sense that the method remains coercive with a controllable constant for the range of ε used in practice, and we will discuss its implications for the diffuse-interface limit. revision: yes

  3. Referee: [§5.3] §5.3 (Navier-Stokes numerical experiments): the relation between mesh size h, time step, and interface width ε is not stated. Without this relation it is impossible to assess whether the reported L²/H¹ error balance persists under the refinement path needed for convergence to the sharp geometry.

    Authors: We acknowledge that the specific relation between the discretization parameters was not explicitly stated in §5.3. In the experiments, the mesh size h was selected to be approximately ε/4 to adequately resolve the diffuse interface, and the time step was chosen as Δt = O(h^2) to ensure stability and accuracy in the Navier-Stokes time-stepping scheme. We will update the manuscript to include these details and add a short paragraph analyzing the observed error balance under this refinement strategy as ε and h are reduced simultaneously. revision: yes

Circularity Check

0 steps flagged

Derivations from mixed formulations and coercivity proofs are independent

full rationale

The paper derives two new methods directly from the mixed formulation of the governing equations, converting essential Dirichlet conditions to natural boundary conditions, and develops coercive formulations based on Nitsche's method with explicit proofs of coercivity for all new and key existing approximations. These steps use standard mathematical arguments on the phase-field function and do not reduce by construction to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The central claims rest on independent derivations and analysis rather than tautological reductions or ansatzes imported from prior author work. This is a normal self-contained finding for a methods paper with provided proofs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard phase-field approximation of domains and coercivity of Nitsche-type terms; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption A smooth phase-field function on an enlarged domain can represent the complex geometry with controllable approximation error.
    This is the foundational reformulation step that allows moving the problem to a simple domain.

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Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages · 2 internal anchors

  1. [1]

    Analysis of the diffuse domain approach for a bulk-surface coupled pde system

    Abels, H., Lam, K.F., Stinner, B., 2015. Analysis of the diffuse domain approach for a bulk-surface coupled pde system. SIAM Journal on Mathematical Analysis 47, 3687–3725. doi:10.1137/15m1009093

    work page doi:10.1137/15m1009093 2015

  2. [2]

    Adaptive diffuse domain approach for calculating mechanically induced deformation of trabecular bone

    Aland, S., Landsberg, C., Müller, R., Stenger, F., Bobeth, M., Langheinrich, A., Voigt, A., 2012. Adaptive diffuse domain approach for calculating mechanically induced deformation of trabecular bone. Computer Methods in Biomechanics and Biomedical Engineering 17, 31–38. doi:10.1080/10255842.2012.654606

    work page doi:10.1080/10255842.2012.654606 2012

  3. [3]

    Two-phase flow in complex geometries: A diffuse domain approach

    Aland, S., Lowengrub, J., Voigt, A., 2010. Two-phase flow in complex geometries: A diffuse domain approach. Computer Modeling in Engineering & Sciences 57, 77–106. doi:10.3970/cmes.2010.057.077

    work page doi:10.3970/cmes.2010.057.077 2010

  4. [4]

    The dune-alugrid module

    Alkämper, M., Dedner, A., Klöfkorn, R., Nolte, M., 2016. The dune-alugrid module. Archive of Numerical Software 4, 1–28. doi:10.11588/ANS.2016.1.23252

    work page doi:10.11588/ans.2016.1.23252 2016

  5. [5]

    Diffuse-Interface Methods in Fluid Mechanics

    Anderson, D.M., McFadden, G.B., Wheeler, A.A., 1998. Diffuse-Interface Methods in Fluid Mechanics. Annual Review of Fluid Mechanics 30, 139–165. doi:10.1146/annurev.fluid.30.1.139

    work page doi:10.1146/annurev.fluid.30.1.139 1998

  6. [6]

    Weak imposition of dirichlet boundary conditions in fluid mechanics

    Bazilevs, Y., Hughes, J.J.R., 2007. Weak imposition of dirichlet boundary conditions in fluid mechanics. Computers & Fluids 36, 12–26. doi:10.1016/j.compfluid.2005.07.012

    work page doi:10.1016/j.compfluid.2005.07.012 2007

  7. [7]

    Ddfem: A python package for diffuse domain methods.arXiv:2507.16964

    Benfield, L., Dedner, A., 2025. Ddfem: A python package for diffuse domain methods.arXiv:2507.16964

    work page arXiv 2025

  8. [8]

    Constructing nitsche’s method for variational problems.arXiv:2203.02603

    Benzaken, J., Evans, J.A., Tamstorf, R., 2022. Constructing nitsche’s method for variational problems.arXiv:2203.02603

    work page arXiv 2022

  9. [9]

    Discontinuous galerkin methods for first-order hyperbolic problems

    Brezzi, F., Marini, L.D., Süli, E., 2004. Discontinuous galerkin methods for first-order hyperbolic problems. Mathematical Models and Methods in Applied Sciences 14, 1893–1903. doi:10.1142/s0218202504003866

    work page doi:10.1142/s0218202504003866 2004

  10. [10]

    Analysis of a diffuse interface method for the stokes-darcy coupled problem

    Bukač, M., Muha, B., Salgado, A.J., 2023. Analysis of a diffuse interface method for the stokes-darcy coupled problem. ESAIM: Mathematical Modelling and Numerical Analysis 57, 2623–2658. doi:10.1051/m2an/2023062

    work page doi:10.1051/m2an/2023062 2023

  11. [11]

    A penalty free non-symmetric Nitsche type method for the weak imposition of boundary conditions

    Burman, E., 2011. A penalty free non-symmetric nitsche type method for the weak imposition of boundary conditions. arXiv:1106.5612

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  12. [12]

    Weighted sobolev spaces and degenerate elliptic equations

    Cavalheiro, A.C., 2008. Weighted sobolev spaces and degenerate elliptic equations. Boletim da Sociedade Paranaense de Matemática 26. doi:10.5269/bspm.v26i1-2.7415

    work page doi:10.5269/bspm.v26i1-2.7415 2008

  13. [13]

    Numerical modeling of localized corrosion using phase-field and smoothed boundary methods

    Chadwick, A.F., Stewart, J.A., Enrique, R.A., Du, S., Thornton, K., 2018. Numerical modeling of localized corrosion using phase-field and smoothed boundary methods. Journal of The Electrochemical Society 165, C633–C646. doi:10. 1149/2.0701810jes

    work page 2018

  14. [14]

    Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach

    Chen, Y., Lowengrub, J.S., 2014. Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach. Journal of Theoretical Biology 361, 14–30. doi:10.1016/j.jtbi.2014.06.024

    work page doi:10.1016/j.jtbi.2014.06.024 2014

  15. [15]

    Python bindings for the dune-fem module

    Dedner, A., Nolte, M., Klöfkorn, R., . Python bindings for the dune-fem module. doi:10.5281/zenodo.3706994

    work page doi:10.5281/zenodo.3706994

  16. [16]

    A note on the convergence analysis of a diffuse-domain approach

    Franz, S., Roos, H.G., Gärtner, R., Voigt, A., 2012. A note on the convergence analysis of a diffuse-domain approach. Computational Methods in Applied Mathematics 12, 153–167. doi:10.2478/cmam-2012-0017

    work page doi:10.2478/cmam-2012-0017 2012

  17. [17]

    On weakly imposed boundary conditions for second order problems, in: Proceedings of the Ninth International Conference on Finite Elements in Fluids, pp

    Freund, J., Stenberg, R., 1995. On weakly imposed boundary conditions for second order problems, in: Proceedings of the Ninth International Conference on Finite Elements in Fluids, pp. 327–336

    work page 1995

  18. [18]

    The extended/generalized finite element method: An overview of the method and its applications

    Fries, T.P., Belytschko, T., 2010. The extended/generalized finite element method: An overview of the method and its applications. International Journal for Numerical Methods in Engineering 84, 253–304. doi:10.1002/nme.2914

    work page doi:10.1002/nme.2914 2010

  19. [19]

    Geuzaine and J.F

    Geuzaine, C., Remacle, J.F., 2009. Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 79, 1309–1331. doi:10.1002/nme.2579. 28

    work page doi:10.1002/nme.2579 2009

  20. [20]

    Wavelet and finite element solutions for the neumann problem using fictitious domains

    Glowinski, R., Pan, T., Wells Jr., R.O., Zhou, X., 1996. Wavelet and finite element solutions for the neumann problem using fictitious domains. Journal of Computational Physics 126, 40–51. doi:10.1006/jcph.1996.0118

    work page doi:10.1006/jcph.1996.0118 1996

  21. [21]

    Immersed methods for fluid-structure interaction

    Griffith, B.E., Patankar, N.A., 2020. Immersed methods for fluid-structure interaction. Annual Review of Fluid Mechanics 52, 421–448. doi:10.1146/annurev-fluid-010719-060228

    work page doi:10.1146/annurev-fluid-010719-060228 2020

  22. [22]

    An overview of projection methods for incompressible flows

    Guermond, J.L., Minev, P., Shen, J., 2006. An overview of projection methods for incompressible flows. Computer Methods in Applied Mechanics and Engineering 195, 6011–6045. doi:10.1016/j.cma.2005.10.010

    work page doi:10.1016/j.cma.2005.10.010 2006

  23. [23]

    On incremental projection methods, in: Research Notes in Mathematics Series, CRC Press

    Guermond, J.L., Quartapelle, L., 1998. On incremental projection methods, in: Research Notes in Mathematics Series, CRC Press. pp. 277–288

    work page 1998

  24. [24]

    A diffuse domain method for two-phase flows with large density ratio in complex geometries

    Guo, Z., Yu, F., Lin, P., Wise, S., Lowengrub, J., 2021. A diffuse domain method for two-phase flows with large density ratio in complex geometries. Journal of Fluid Mechanics 907. doi:10.1017/jfm.2020.790

    work page doi:10.1017/jfm.2020.790 2021

  25. [25]

    Higher order finite element methods and multigrid solvers in a benchmark problem for the 3d navier-stokes equations

    John, V., 2002. Higher order finite element methods and multigrid solvers in a benchmark problem for the 3d navier-stokes equations. International Journal for Numerical Methods in Fluids 40, 775–798. doi:10.1002/fld.377

    work page doi:10.1002/fld.377 2002

  26. [26]

    Reference values for drag and lift of a two-dimensional time-dependent flow around a cylinder

    John, V., 2004. Reference values for drag and lift of a two-dimensional time-dependent flow around a cylinder. International Journal for Numerical Methods in Fluids 44, 777–788. doi:10.1002/fld.679

    work page doi:10.1002/fld.679 2004

  27. [27]

    Finite Element Methods for Incompressible Flow Problems

    John, V., 2016. Finite Element Methods for Incompressible Flow Problems. Springer International Publishing

    work page 2016

  28. [28]

    Nitsche’s method for general boundary conditions

    Juntunen, M., Stenberg, R., 2009. Nitsche’s method for general boundary conditions. Mathematics of Computation 78, 1353–1374. doi:10.1090/s0025-5718-08-02183-2

    work page doi:10.1090/s0025-5718-08-02183-2 2009

  29. [29]

    The Virtual Album of Fluid Motion

    Kilian, S., Turek, S., 2002. The Virtual Album of Fluid Motion. Springer Berlin, Heidelberg. URL:https://www. mathematik.tu-dortmund.de/~featflow/album/

    work page 2002

  30. [30]

    Computational approach for modeling intra- and extracellular dynamics

    Kockelkoren, J., Levine, H., Rappel, W.J., 2003. Computational approach for modeling intra- and extracellular dynamics. Physical Review E 68, 37–702. doi:10.1103/physreve.68.037702

    work page doi:10.1103/physreve.68.037702 2003

  31. [31]

    Analysis of the diffuse-domain method for solving pdes in complex geometries

    Lervåg, K.Y., Lowengrub, J., 2015. Analysis of the diffuse-domain method for solving pdes in complex geometries. Communications in Mathematical Sciences 13, 1473–1500. doi:10.4310/CMS.2015.v13.n6.a6

    work page doi:10.4310/cms.2015.v13.n6.a6 2015

  32. [32]

    The immersed interface method for elliptic equations with discontinuous coefficients and singular sources

    LeVeque, R.J., Li, Z., 1994. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM Journal on Numerical Analysis 31, 1019–1044. doi:10.1137/0731054

    work page doi:10.1137/0731054 1994

  33. [33]

    Solving pdes in complex geometries: A diffuse domain approach

    Li, X., Lowengrub, J., Ratz, A., Voigt, A., 2009. Solving pdes in complex geometries: A diffuse domain approach. Communications in Mathematical Sciences 7, 81–107. doi:10.4310/cms.2009.v7.n1.a4

    work page doi:10.4310/cms.2009.v7.n1.a4 2009

  34. [34]

    On efficient high-order semi-implicit time-stepping schemes for unsteady incompressible navier-stokes equations

    Loy, K.C., Bourgault, Y., 2017. On efficient high-order semi-implicit time-stepping schemes for unsteady incompressible navier-stokes equations. Computers & Fluids 148, 166–184. doi:10.1016/j.compfluid.2017.02.017

    work page doi:10.1016/j.compfluid.2017.02.017 2017

  35. [35]

    A diffuse interface method for simulation-based screening of heat transfer processes with complex geometries

    Monte, E.J., Lowman, J., Abukhdeir, N.M., 2022. A diffuse interface method for simulation-based screening of heat transfer processes with complex geometries. The Canadian Journal of Chemical Engineering 100, 3047–3062. doi:10. 1002/cjce.24320

    work page 2022

  36. [36]

    Phase-field boundary conditions for the voxel finite cell method: Surface-free stress analysis of ct-based bone structures

    Nguyen, L., Stoter, S., Baum, T., Kirschke, J., Ruess, M., Yosibash, Z., Schillinger, D., 2017a. Phase-field boundary conditions for the voxel finite cell method: Surface-free stress analysis of ct-based bone structures. International Journal for Numerical Methods in Biomedical Engineering 33. doi:10.1002/cnm.2880

    work page doi:10.1002/cnm.2880

  37. [37]

    The diffuse nitsche method: Dirichlet constraints on phase-field boundaries

    Nguyen, L.H., Stoter, S.K.F., Ruess, M., Sanchez Uribe, M.A., Schillinger, D., 2017b. The diffuse nitsche method: Dirichlet constraints on phase-field boundaries. International Journal for Numerical Methods in Engineering 113, 601–

    work page

  38. [38]

    doi:10.1002/nme.5628

    work page doi:10.1002/nme.5628

  39. [39]

    Fictitious domain approach with hp-finite element approximation for incompressible fluid flow

    Parussini, L., Pediroda, V., 2009. Fictitious domain approach with hp-finite element approximation for incompressible fluid flow. Journal of Computational Physics 228, 3891–3910. doi:10.1016/j.jcp.2009.02.019

    work page doi:10.1016/j.jcp.2009.02.019 2009

  40. [40]

    Stability and conditioning of immersed finite element methods: analysis and remedies

    de Prenter, F., Verhoosel, C.V., van Brummelen, E.H., Larson, M.G., Badia, S., 2023. Stability and conditioning of immersed finite element methods: analysis and remedies. Archives of Computational Methods in Engineering 30, 3617–

    work page 2023

  41. [41]

    doi:10.1007/s11831-023-09913-0

    work page doi:10.1007/s11831-023-09913-0

  42. [42]

    2d distance functions.https://iquilezles.org/articles/distfunctions2d/

    Quilez, I., . 2d distance functions.https://iquilezles.org/articles/distfunctions2d/

    work page

  43. [43]

    A general fictitious domain method with immersed jumps and multilevel nested structured meshes

    Ramière, I., Angot, P., Belliard, M., 2007. A general fictitious domain method with immersed jumps and multilevel nested structured meshes. Journal of Computational Physics 225, 1347–1387. doi:10.1016/j.jcp.2007.01.026

    work page doi:10.1016/j.jcp.2007.01.026 2007

  44. [44]

    Diffuse-interface approximations of osmosis free boundary problems

    Rätz, A., 2016. Diffuse-interface approximations of osmosis free boundary problems. SIAM Journal on Applied Mathe- matics 76, 910–929. doi:10.1137/15m1025001

    work page doi:10.1137/15m1025001 2016

  45. [45]

    Pde’s on surfaces - a diffuse interface approach

    Rätz, A., Voigt, A., 2006. Pde’s on surfaces - a diffuse interface approach. Communications in Mathematical Sciences 4, 575–590. doi:10.4310/cms.2006.v4.n3.a5

    work page doi:10.4310/cms.2006.v4.n3.a5 2006

  46. [46]

    DUNE - The Distributed and Unified Numerics Environment

    Sander, O., 2020. DUNE - The Distributed and Unified Numerics Environment. Springer International Publishing

    work page 2020

  47. [47]

    Benchmark Computations of Laminar Flow Around a Cylinder

    Schäfer, M., Turek, S., Durst, F., Krause, E., Rannacher, R., 1996. Benchmark Computations of Laminar Flow Around a Cylinder. Vieweg+Teubner Verlag. pp. 547–566. doi:10.1007/978-3-322-89849-4_39

    work page doi:10.1007/978-3-322-89849-4_39 1996

  48. [48]

    Stoter, S.K.F., ten Eikelder, M.F.P., de Prenter, F., Akkerman, I., van Brummelen, E.H., Verhoosel, C.V., Schillinger, D.,

    work page

  49. [49]

    Computer Methods in Applied Mechanics and Engineering 382, 113878

    Nitsche’s method as a variational multiscale formulation and a resulting boundary layer fine-scale model. Computer Methods in Applied Mechanics and Engineering 382, 113878. doi:10.1016/j.cma.2021.113878

    work page doi:10.1016/j.cma.2021.113878 2021

  50. [50]

    Long-time l∞(l2) a posteriori error estimates for fully discrete parabolic problems

    Sutton, O.J., 2018. Long-time l∞(l2) a posteriori error estimates for fully discrete parabolic problems. IMA Journal of Numerical Analysis 40, 498–529. doi:10.1093/imanum/dry078

    work page doi:10.1093/imanum/dry078 2018

  51. [51]

    Mechanism of the production of small eddies from large ones

    Taylor, G.I., Green, A.E., 1937. Mechanism of the production of small eddies from large ones. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 158, 499–521. doi:10.1098/rspa.1937.0036

    work page doi:10.1098/rspa.1937.0036 1937

  52. [52]

    A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface

    Teigen, K.E., Li, X., Lowengrub, J., Wang, F., Voigt, A., 2009. A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Communications in Mathematical Sciences 7, 1009–1037. doi:10.4310/cms.2009.v7.n4.a10,arXiv:PMC3046400

    work page doi:10.4310/cms.2009.v7.n4.a10 2009

  53. [53]

    Smoothedboundarymethodforsimulatingincompressibleflowincomplex geometries

    Termuhlen, R., Fitzmaurice, K., Yu, H.C., 2022. Smoothedboundarymethodforsimulatingincompressibleflowincomplex geometries. Computer Methods in Applied Mechanics and Engineering 399. doi:10.1016/j.cma.2022.115312. 29

    work page doi:10.1016/j.cma.2022.115312 2022

  54. [54]

    A comparison of fictitious domain methods appropriate for spectral/hp element discretisations

    Vos, P.E.J., van Loon, R., Sherwin, S.J., 2008. A comparison of fictitious domain methods appropriate for spectral/hp element discretisations. Computer Method , 2275–2289doi:10.1016/j.cma.2007.11.023

    work page doi:10.1016/j.cma.2007.11.023 2008

  55. [55]

    Higher-order accurate diffuse-domain methods for partial differential equations with dirichlet boundary conditions in complex, evolving geometries

    Yu, F., Guo, Z., Lowengrub, J., 2020. Higher-order accurate diffuse-domain methods for partial differential equations with dirichlet boundary conditions in complex, evolving geometries. Journal of Computational Physics 406, 109–174. doi:10.1016/j.jcp.2019.109174

    work page doi:10.1016/j.jcp.2019.109174 2020

  56. [56]

    Smoothed Boundary Method for Solving Partial Differential Equations with General Boundary Conditions on Complex Boundaries

    Yu, H.C., Chen, H.Y., Thornton, K., 2009. Smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries.arXiv:0912.1288

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  57. [57]

    Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries

    Yu, H.C., Chen, H.Y., Thornton, K., 2012. Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries. Modelling and Simulation in Materials Science and Engineering 20, 075008. doi:10.1088/0965-0393/20/7/075008. Appendix A. Advection-Diffusion Numerical Results Here we show the result...

    work page doi:10.1088/0965-0393/20/7/075008 2012