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Parametric finite element approximations of curvature driven interface evolutions

John W. Barrett, Harald Garcke, Robert Nürnberg

arXiv:1903.09462v3math.NAcs.NA

TL;DR

Curvature-driven interface evolutions require numerical methods for geometric motion and for coupling interface laws to bulk equations. The paper develops parametric finite element techniques, including weak formulations and unfitted meshes. It reports stable schemes and favorable mesh behavior, with a limitation for one Willmore-flow approximation.

  • Problem

    Curvature-driven interface laws, especially those coupled to bulk partial differential equations, pose challenging computational problems, and evolving polyhedral meshes can become highly inhomogeneous.

    Source: §4.6 · p. 46

  • Method

    The paper develops parametric finite element methods based on weak formulations for mean curvature, including unfitted methods with a surface mesh independent of the bulk mesh.

    Source: §1 · p. 2, §4.1 · p. 41

  • Results

    The schemes are reported stable for strongly anisotropic, nearly crystalline anisotropies, while particular formulations yield equidistribution in two dimensions and conformal polyhedral hypersurfaces in three.

    Source: §6.3 · p. 70, §8.1.1 · p. 89

  • Takeaways & Limitations

    The discussed approach typically maintains well-behaved meshes, avoiding close mesh points and extreme angles that can make computations unstable or inaccurate.

    Source: §4.6 · p. 46

    The Willmore-flow approximation (178) lacks good mesh properties because its mesh movements are almost exclusively in the normal direction.

    Source: §9.3 · p. 103

Abstract

from arXiv · show

Parametric finite elements lead to very efficient numerical methods for surface evolution equations. We introduce several computational techniques for curvature driven evolution equations based on a weak formulation for the mean curvature. The approaches discussed, in contrast to many other methods, have good mesh properties that avoid mesh coalescence and very non-uniform meshes. Mean curvature flow, surface diffusion, anisotropic geometric flows, solidification, two-phase flow, Willmore and Helfrich flow as well as biomembranes are treated. We show stability results as well as results explaining the good mesh properties.

1 Introduction

Curvature-driven interface evolution arises when surface energies depend on area or curvature, while bulk-coupled laws add computational challenges. The paper develops parametric finite element methods for these evolutions, including unfitted methods for interface–bulk problems.

1 Introduction

  • Area-driven evolution leads naturally to mean curvature flow, while curvature-dependent energies such as the Willmore functional lead to other challenging flows.The Willmore functional is the integrated squared mean curvature.Source: §1 · p. 2
  • Bulk-coupled interface laws, including solidification with temperature governed by diffusion, pose more involved analysis and computation problems.Interface evolution may also couple to flow fields governed by Stokes or Navier–Stokes systems.Source: §1 · p. 2
  • The paper presents parametric finite element methods for evolving interfaces, including unfitted methods for coupling interface equations to bulk equations.Its examples include solidification and two-phase fluid flow, as well as Willmore flow and biomembranes.Source: §1 · p. 2

2 Geometry of surfaces

This section reviews the surface concepts needed to develop numerical algorithms for curvature-driven flows of hypersurfaces.

2 Geometry of surfaces

  • The section reviews basic facts about surfaces needed for later numerical algorithms for curvature-driven hypersurface flows.It covers surface differential operators, curvature quantities, and a divergence theorem on hypersurfaces.Source: §2 · p. 2

2.1 Surfaces in Rd

The section establishes surface geometry and differential-operator notation used in the numerical analysis. It also gives representations and extension identities that support calculations on hypersurfaces.

2.1 Surfaces in Rd

  • A Ck surface is locally parameterized by a full-rank map, whose derivatives span the tangent space; orientability requires a continuous unit normal field.The normal space is the orthogonal complement of the tangent space.Source: §2.1 · p. 3
  • The section defines surface differential operators, including the surface gradient, Laplace–Beltrami operator, generalized symmetrized surface Jacobian, and surface divergence.The vector Laplace–Beltrami operator is defined component-wise.Source: §2.1 · pp. 3–4

(iii)

  • Surface derivatives are independent of the chosen curve parameterization or tangent-space orthonormal basis, and the surface gradient is tangential.The surface gradient and divergence can also be represented using ambient-space derivatives and the unit normal.Source: §2.1 · p. 5, (iii) · p. 7
  • Local parameterizations give the first fundamental form and a representation of the Laplace–Beltrami operator, while the surface Jacobian projections enforce tangentiality and symmetry.The first fundamental form determines the square of the local area element.Source: §2.1 · p. 5, (iii) · pp. 6–7

2.2 Curvature

This section defines curvature quantities and records identities used in the analysis of curvature-driven flows. It also develops surface extensions and differentiation formulas for later calculations.

2.2 Curvature

  • Principal curvatures are eigenvalues of the Weingarten map, mean curvature is their sum, and in three dimensions Gaussian curvature is their product.The mean curvature is also the trace of the Weingarten map.Source: §2.2 · p. 8
  • The surface derivative of the unit normal induces the negative Weingarten map on tangent vectors, and the mean curvature vector is invariant under reversal of the normal.The sign of scalar mean curvature depends on the choice of unit normal.Source: §2.2 · p. 8, (iii) · p. 9

(iii)

  • For a compact orientable hypersurface, functions can be extended constantly along normals to a sufficiently small tubular neighborhood, making ambient and surface gradients agree on the surface.The neighborhood projection and signed distance provide the associated projection and distance functions.Source: (iii) · pp. 10–11
  • On curved hypersurfaces, the surface Hessian need not be symmetric; its antisymmetric part is expressed using the surface derivative of the normal and the surface gradient.The section also gives a decomposition of a vector-field surface derivative into normal and tangential parts.Source: (iii) · pp. 9, 11

2.3 The divergence theorem

This section develops surface integration and divergence identities for hypersurfaces, including a divergence theorem that accounts for the normal component of a vector field. It also highlights identities useful in numerical approximations of curvature-driven evolution equations.

2.3 The divergence theorem

  • For surfaces with boundary, the boundary is an (n−1)-dimensional surface, and its outer unit conormal lies in the surface tangent space.The conormal points into the surface along a curve through the boundary point.Source: §2.3 · pp. 12–13
  • For a non-tangential vector field, the surface divergence theorem incorporates a curvature-weighted normal component, while its boundary flux is the field’s conormal component.The proof removes the normal component to form a tangential field, then applies the tangential divergence theorem.Source: §2.3 · pp. 13–14
  • The section extends integration by parts identities to surfaces with boundary using the surface divergence theorem.It gives scalar and vector versions, including a form involving symmetric matrix fields.Source: §2.3 · p. 14
  • The section identifies a surface identity as fundamental to developing numerical approximations of curvature-driven evolution equations.It also presents related identities for approximating the Weingarten map.Source: §2.3 · p. 14

2.4 Evolving surfaces and transport theorems

This section defines evolving hypersurfaces, their velocities, and time derivatives, then derives transport results for moving surfaces and domains. The framework distinguishes geometric normal motion from parameterization-dependent tangential motion.

2.4 Evolving surfaces and transport theorems

  • The normal velocity is independent of the curve used to define it, while the full velocity’s normal component is parameterization-independent and its tangential component depends on the parameterization.The section also establishes orientability of the evolving hypersurface.Source: §2.4 · pp. 15–16
  • The time derivative on an evolving surface requires moving the point as time changes; the text defines both a parameterization-induced material derivative and a normal time derivative.The normal time derivative follows motion with velocity Vν.Source: §2.4 · pp. 17–18
  • The area element evolves according to the surface divergence of the velocity, equivalently −Vκ plus the tangential velocity’s surface divergence.This formula is used in deriving transport results for evolving hypersurfaces.Source: §2.4 · p. 19
  • The section establishes transport theorems for compact evolving hypersurfaces and for moving domains bounded by them.It also extends the domain result to two-phase moving domains with separate phase functions.Source: §2.4 · pp. 19, 20, 22
  • For two-phase domains, the interface separates Ω−(t) from Ω+(t), and the transport theorem applies to functions specified separately in each phase.The phase functions are assumed to be C1 on their respective domains.Source: §2.4 · pp. 21–22

2.5 Time derivatives of the normal

The section derives the normal time derivative of the unit normal on an evolving orientable hypersurface. It expresses this derivative through the surface gradient of the normal velocity.

2.5 Time derivatives of the normal

  • For a C2-evolving orientable hypersurface, the normal time derivative of the unit normal equals the negative surface gradient of the normal velocity.The proof uses the orthogonality of the normal to a basis of tangent vectors.Source: §2.5 · pp. 22–23

2.6 Time derivatives of the mean curvature

Time derivatives of curvature are needed to compute first variations of energies that depend on mean curvature. The section gives commutator rules and formulas for the time derivatives of mean curvature.

2.6 Time derivatives of the mean curvature

  • A commutator rule for time derivatives and surface differential operators is introduced as part of the curvature-derivative calculations.These curvature derivatives support first variations of energies depending on mean curvature.Source: §2.6 · p. 23, (iii) · p. 24

(iii)

  • The full time derivative of mean curvature adds tangential advection to the normal time derivative formula.The advection term is the tangential velocity dotted with the surface gradient of mean curvature.Source: (iii) · p. 24
  • The normal time derivative of mean curvature is the surface Laplacian of normal velocity plus normal velocity times the squared norm of the surface gradient of the normal.This is the curvature derivative used without tangential transport.Source: (iii) · p. 24

2.7 Gauss–Bonnet theorem

For d = 3, the section introduces curvature vectors for curves on a hypersurface and invokes Gauss–Bonnet for the special case of the hypersurface boundary.

2.7 Gauss–Bonnet theorem

  • For d = 3, curvature vectors can be defined for curves on a hypersurface.Source: §2.7 · p. 25
  • The second derivative with respect to arclength on γ is invariant under a change of curve parameterization.Source: §2.7 · p. 25
  • For a compact orientable C2 hypersurface in R3, the Gauss–Bonnet statement involves its Euler characteristic m(Γ).The theorem is used for the special case g = ∂Γ.Source: §2.7 · p. 25

3 Parametric finite elements

This section sets out polyhedral and finite-element concepts for approximating curvature-driven evolution, then develops stability estimates for discrete surface maps.

3.1 Polyhedral surfaces

  • A polyhedral surface is assembled from nondegenerate simplices whose intersections are empty or shared lower-dimensional simplices.Polyhedral surfaces are used to approximate smooth surfaces.Source: §3.1 · p. 26

3.1.1 Orientation

  • Consistent simplex orientations define a piecewise-constant unit normal, whose mass-lumped L2 projection gives the vertex normal.Source: §3.1.1 · pp. 28–29

3.1.2 Polygonal curves

  • A closed polygonal curve is parameterized on the periodic unit interval by a continuous piecewise-linear finite-element function.Source: §3.1.2 · pp. 29–30

3.2 Stability estimates

  • The estimate in Lemma 55 applies only to 2-dimensional polyhedral surfaces; it does not extend to n ≠ 2.Source: §3.2 · pp. 31–32
  • A separate stability bound is established for n = 1 and n = 2 polyhedral surfaces.The passage identifies this as a crucial bound used extensively in later sections.Source: §3.2 · p. 32
  • The fully implicit analogue is treated for polygonal curves; an analogous Lemma 58 result is false for n > 1 surfaces.Source: §3.2 · p. 33

3.3 Curvature approximations

The section develops discrete curvature approximations and establishes geometric properties for special and general polyhedral surfaces. It then sets up evolving-surface transport results and states their discrete analogues.

3.3 Curvature approximations

  • For closed conformal polygonal curves, neighboring elements have equal lengths or are parallel.The proof notes that nonparallel neighboring elements must have equal lengths.Source: §3.3 · pp. 34–35
  • For general polyhedral hypersurfaces, system (33) has a unique solution under a mild assumption, and its identity-map solution yields a conformal surface.The uniqueness result is stated when Assumption 64 holds.Source: §3.3 · p. 35
  • Curvature approximation (29) is O(h)-accurate in [L2(I)]2 for smooth curves, but (29) and (35a–c) are not convergent on general 3D meshes.The passage says (35d) and (33) behave better in practice; higher-order piecewise polynomial approximations admit provable convergence.Source: §3.3 · p. 37

3.4 Evolving polyhedral surfaces and transport theorems

  • Section 3.4 defines evolving polyhedral surfaces, their velocity fields, and material time derivatives as a basis for discrete transport results.The section introduces discrete analogues of evolving hypersurfaces, velocity fields, and material derivatives.Source: §3.4 · pp. 37–38

3.5 Further results for evolving polyhedral surfaces

  • Section 3.5 states discrete analogues of earlier results for evolving polyhedral surfaces, including identities involving the surface velocity gradient.The stated results define the tangential symmetric part of the velocity gradient and relate it to surface-gradient evolution.Source: §3.5 · pp. 39–40

(iii)

  • A further identity expresses the material derivative of a surface gradient through the velocity gradient and the tangential symmetric velocity gradient.The section gives the corresponding contraction identity as well.Source: §3.5 · p. 40, (iii) · p. 40

4 Mean curvature flow

The section presents numerical schemes for mean curvature flow, including weak-form parametric finite element methods for closed hypersurfaces and curve-flow schemes. It considers their stability, mesh behavior and equidistribution, and convergence guarantees, noting scheme-specific qualifications.

4.1 Weak formulation

  • The method approximates V = κ through a weak formulation of κν = Δ_s id, forming the basis for stable approximations with generally good mesh quality.The formulation is for closed hypersurfaces and uses a global parameterization.Source: §4.1 · pp. 40–41

4.2 Finite element approximation

  • The time-stepping finite element approximation uses the current polyhedral surface geometry explicitly, making the resulting system linear despite the nonlinear evolution problem.The approach can be adapted to boundary conditions, including prescribed contact angles and attached surface clusters.Source: §4.2 · pp. 41–42

4.3 Discrete linear systems

  • The discrete equations can be assembled as a linear system and solved efficiently with sparse direct methods or a Schur-complement-based preconditioned conjugate-gradient approach.The section also presents a formulation with the updated parameterization as the only unknown.Source: §4.3 · pp. 43–44

4.3.1 Curves in the plane

  • For closed curves in the plane, the finite element scheme can be rewritten as a finite difference-type system for nodal positions and curvature.The resulting equations can also be reduced to an equation for the updated nodal positions alone.Source: §4.3.1 · pp. 44–45

4.4 Existence and uniqueness

  • Under Assumption 64(i), the finite element system (41) has a unique solution at each time step.The section notes that existence follows from uniqueness for this system.Source: §4.4 · p. 45

4.5 Stability

  • The fully discrete scheme is unconditionally stable for d = 2 and d = 3.The section also derives an energy bound over successive time steps.Source: §4.5 · pp. 45–46

4.6 Equipartition property

  • The method’s tangential motion typically preserves good mesh quality, addressing coalescence and highly non-uniform meshes that can destabilize computations or prevent them from continuing.The semidiscrete scheme is stable and yields conformal polyhedral surfaces; a fully implicit discretization also produces conformal surfaces at every time step. Solutions to these semidiscrete and fully implicit schemes are not assured in general, and the fully discrete scheme (41) does not inherit conformality.Source: §4.6 · pp. 46–48
  • The adapted scheme (63) fully equidistributes curve vertices at every time step, while also satisfying the stability estimate inherited from (62).This sharpens the earlier result, which guaranteed equidistribution only where curve elements were not locally parallel.Source: §4.6 · pp. 49–50

4.7.1 The classical Dziuk approach

  • The classical Dziuk approach is a linear, stable method; for d = 2, a semidiscrete error estimate is available under a smoothness assumption, but no d = 3 convergence proof is given.In practice, both this approximation and scheme (41) appear convergent.Source: §4.7.1 · p. 51

4.7.2 The convergent finite element algorithm of Kov´acs, Li, Lubich

  • Kovács et al. (2019) give the first convergence proof for mean curvature flow in d = 3, with error estimates for semidiscrete and fully discrete schemes under smoothness assumptions.Their approach can still produce deteriorating meshes and requires an additional system, but is the only method described here with a d = 3 convergence proof.Source: §4.7.2 · p. 51

4.7.3 Alternative numerical methods that equidistribute

  • Alternative curve methods control mesh-point distribution through arclength parameterizations or tangential velocities, while mesh smoothing can sacrifice stability or conservation.The discussed approaches include methods that enforce equidistribution; a fully discrete method using a side constraint can lead to very nonlinear problems.Source: §4.7.3 · pp. 51–52

4.7.4 Using the DeTurck trick to obtain good mesh properties

  • The DeTurck-based method of Elliott and Fritz (2017) can preserve good meshes from a good initial triangulation, but uses more complicated systems and is less flexible.It may produce better meshes than the method in §4.2 for d = 3, while being less suited to anisotropy, triple junctions, and coupling to other fields.Source: §4.7.4 · p. 53

4.7.5 Other numerical approaches

  • Other approaches include graph-based and axisymmetric finite elements, tangential redistribution on evolving surfaces, and level set and phase field methods.The cited tangential redistribution methods use volume-oriented and length-oriented strategies applicable in any dimension.Source: §4.7.5 · p. 53

5 Surface diffusion and other flows

This section extends parametric finite element schemes to surface diffusion and other hypersurface flows, examining tangential motion and discrete stability and conservation properties. It also contrasts an alternative surface-diffusion scheme with generally deteriorating meshes and notes that normalized-normal variants do not preserve volume.

5.1 Properties of the surface diffusion flow

  • Surface diffusion preserves enclosed volume while decreasing surface area for closed surfaces.The flow can also be interpreted as the H−1-gradient flow of surface area.Source: §5.1 · p. 54

5.2 Finite element approximation for surface diffusion

  • The fully discrete surface-diffusion scheme has a unique solution and is unconditionally stable in dimensions 2 and 3.Uniqueness holds when the current polyhedral hypersurface satisfies Assumption 64(i).Source: §5.2 · pp. 54–55

5.3 Volume conservation for the semidiscrete scheme

  • The semidiscrete scheme mimics the surface-diffusion energy law and conserves enclosed volume; its evolving polyhedral surfaces are conformal.The fully discrete scheme does not conserve volume exactly, but its relative enclosed-volume loss tends to zero as the time step decreases.Source: §5.3 · pp. 56–57

5.4 Generalizations to other flows

  • For flows whose operator satisfies the stated energy condition, the generalized finite element scheme has a stability result and a unique solution in the linear case.The framework includes mean curvature flow and surface diffusion; it also yields a stable discretization for a flow interpolating between SALK and surface diffusion.Source: §5.4 · pp. 57–59
  • The generalized framework also covers nonlinear curvature laws and volume-conserving flows, with stability results for the resulting schemes.For curves, fully implicit variants inherit stability and satisfy the strong equidistribution property.Source: §5.4 · pp. 58–60

5.5 Approximations with reduced or induced tangential motion

  • An alternative formulation can reduce or induce tangential motion in selected directions, and the adapted linear system has a unique solution.The formulation augments the unknowns with tangential-motion variables.Source: §5.5 · pp. 60–61
  • For d = 2 or 3, under the stated conditions, (91) provides a stability estimate for scheme (90).The stated condition includes c_i = 0 for i = 1, …, d − 1.Source: §5.5 · p. 61
  • Tangential-motion strategies can reduce motion, steer vertices toward neighboring-node barycentres, or fix their tangential components.Increasing α in (S1) reduces tangential motion; (S2) induces movement toward barycentres, while (S3) fixes tangential motion after each time step.Source: §5.5 · p. 62
  • A semidiscrete variant of scheme (90) satisfies the analogue of Theorem 88(i), and also Theorem 88(ii) when the approximated flow is volume preserving..Source: §5.5 · p. 62
  • Normalizing the discrete normal preserves most stated results, but generally removes volume preservation, which requires the unnormalized normal in schemes (85) and (90).The passage says all results in Theorems 90 and 93 still hold for the variants, as do most results in Remarks 91(i) and 94.Source: §5.5 · p. 62

5.6 Alternative parametric methods

  • The alternative surface-diffusion scheme of Bänsch et al. uses two more variables than (76), and its meshes generally deteriorate because motion is only normal.The passage contrasts its normal-collinear surface motion with (76).Source: §5.6 · pp. 63–64
  • Other cited work considers semidiscrete surface-diffusion error analyses under smooth-solution assumptions, graph formulations, and an axisymmetric parametric approximation.The passage names Bänsch et al. (2004), Coleman et al. (1996), Deckelnick et al. (2003), and Barrett et al. (2019b).Source: §5.6 · p. 64

6 Anisotropic flows

This section develops finite element approaches for anisotropic mean-curvature flow and surface diffusion, including schemes for strongly anisotropic, nearly crystalline energies. It also proposes curvature-energy regularization for strongly anisotropic surface energies, yielding higher-order flows resembling Willmore flow.

6.1 Derivation of the governing equations

  • Anisotropic mean-curvature flow and surface diffusion use weighted mean curvature κγ; the schemes also stably approximate crystalline energies with flat parts, edges, and corners.The flow laws include a smooth kinetic coefficient β(ν).Source: §6.1 · pp. 64–65

6.2 Suitable weak formulations

  • An SPD-matrix inner product restores an anisotropic curvature identity, and norms of ellipsoidal anisotropies model relevant energies including hexagonal and cubic anisotropies.The resulting weak formulations involve only first derivatives, allowing continuous piecewise linear finite elements.Source: §6.2 · pp. 66–68

6.3 Finite element approximation

  • For r=1, the discrete systems have a unique solution; for r>1, existence and uniqueness are unknown, though iterative solvers converged in practice.The r=1 systems are linear.Source: §6.3 · pp. 69–70
  • Stability results enable the schemes to compute strongly anisotropic, nearly crystalline flows, which are difficult for other schemes.The analysis establishes a local inequality for the anisotropic energy.Source: §6.3 · p. 70

6.4 Solution method and discrete systems

  • For r=1, the discrete system is linear and can be solved with a sparse direct solver or a Schur-complement approach; for r>1, a lagged fixed-point iteration is used.The text reports that this iteration always converged in practice.Source: §6.4 · p. 71

6.5 Volume conservation for semidiscrete schemes

  • Fully discrete schemes produce good-quality meshes without coalescence in practice for the stated anisotropies in d=2 and d=3.Semidiscrete schemes retain anisotropic analogues of volume-conservation results, although their tangential motion generally does not satisfy the cited mesh condition.Source: §6.5 · pp. 71–72

6.6 Alternative numerical approaches

  • Related work covers anisotropic flow schemes, error analyses, extensions to clusters and higher codimension, and level-set and phase-field approaches.Examples include Dziuk (1999a), Barrett et al. (2011), Barrett et al. (2008c, 2010a,d), Pozzi (2007, 2008), Deckelnick and Dziuk (1999), and Deckelnick et al. (2005a, 2005b); level-set and phase-field work includes Clarenz et al. (2005), Burger et al. (2007), Caginalp and Lin (1987), Garcke et al. (1999), Beneˇs (2003), Gr¨aser et al. (2013), and Barrett et al. (2013b, 2014b).Source: §6.6 · p. 72
  • For strongly anisotropic surface energies, curvature-energy regularization yields higher-order flows with similarities to Willmore flow (Burger, 2005; Haußer and Voigt, 2005; Torabi et al., 2009).Source: §6.6 · p. 72

7 Coupling bulk equations to geometric equations on the surface and applications to crystal growth

This section develops finite element approaches for coupling bulk equations to curvature-driven interface motion, covering Mullins–Sekerka and Stefan models as well as one-sided free-boundary problems relevant to snow crystal growth. It also presents stability and snow-crystal simulations for a one-sided scheme and surveys other numerical approaches to crystal growth.

7.1 The Mullins–Sekerka problem

  • The Mullins–Sekerka model couples harmonic bulk fields in both phases to interface motion, with zero normal flux on the domain boundary.The interface velocity is coupled to the jump in normal derivative across the interface.Source: §7.1 · p. 73

7.1.1 Weak formulation of the Mullins–Sekerka problem

  • For sufficiently smooth solutions, the Mullins–Sekerka weak formulation recovers the problem’s bulk and interface conditions and yields decreasing interface area.The weak formulation is shown to imply the strong conditions, and the area decrease follows from the transport theorem and the weak equations.Source: §7.1.1 · pp. 73–74

7.1.2 An unfitted finite element approximation of the Mullins–Sekerka problem

  • The unfitted Mullins–Sekerka scheme has a unique solution and satisfies a stability estimate in d = 2 or 3.Its surface mesh is independent of the bulk mesh. The fully discrete scheme generally lacks exact volume conservation, while a semidiscrete version has exact volume conservation and the stated mesh property.Source: §7.1.2 · pp. 75–76

7.2 The Stefan problem with a (kinetic) Gibbs–Thomson law

  • For the anisotropic Stefan problem, the finite element scheme has a unique solution for r = 1 and satisfies a stability estimate for d = 2 or 3 and r ∈ [1, ∞).The scheme’s energy bound closely mimics the continuous energy law; a semidiscrete version has an exact discrete analogue. When K+ ≠ K−, computing the bulk conductivity term requires phase-region intersections, which can be computationally expensive for an unfitted mesh.Source: §7.2 · pp. 79–80

7.3 One-sided free boundary problems

  • The one-sided free-boundary model restricts diffusion to one phase, a setting relevant to snow crystal growth, and supports anisotropic hexagonal-prism surface energy.The scheme’s existence and uniqueness are shown for r = 1, with stability for r ∈ [1, ∞); when ϑ > 0, it is meaningful only if the discrete solid region does not shrink.Source: §7.3 · pp. 80–82
  • For d = 2 and d = 3, the stability proof for r = 1 carries over to r > 1 using Lemma 102.The cited proofs apply under appropriate assumptions (Barrett et al., 2013a, Theorems 3.1 and 3.2).Source: §7.3 · p. 82
  • The scheme (132) supports two snow-crystal growth simulations with hexagonal surface energy density γ and time-dependent mobility β.Figure 5 presents the simulations; its caption identifies snapshots from 2D and 3D experiments.Source: §7.3 · p. 83, Figure 5 · p. 83

7.4 Alternative numerical approaches

  • Alternative crystal-growth computations include front tracking (Roosen and Taylor, 1991), level sets (Sethian and Strain, 1992), phase fields (Kobayashi, 1993), and cellular automata (Reiter, 2005).Source: §7.4 · p. 83

8 Two-phase flow

This section examines parametric finite-element approaches to two-phase flow with interfaces moving with the fluid, including Stokes formulations and an extension to Navier–Stokes flow. It covers phase-volume behavior, stability and interface mesh quality, spurious velocities, fluidic tangential velocity, alternative interface-tracking methods, and surfactant extensions.

8.1.3 XFEMΓ for conservation of the phase volumes

  • A simple XFEM pressure-space enrichment gives exact volume conservation for the two phase regions.The section identifies phase-volume conservation as a desired discrete property and notes that the semidiscrete scheme conserves the minus-phase volume.Source: §8 · p. 84, §8.1.3 · p. 92

8.1 Two-phase Stokes flow

  • Without external forcing, the two-phase Stokes formulation decreases interface area; leaving tangential interface velocity unspecified leads to discrete schemes with good mesh properties.The interface normal velocity matches the fluid normal velocity, and curvature is represented through the weak formulation.Source: §8.1 · pp. 85–86

8.1.1 Finite element approximation

  • The fully discrete unfitted finite element schemes have unique solutions under stated assumptions and are unconditionally stable.The velocity-pressure spaces are subject to an LBB inf-sup condition for the scheme with pressure; the reduced scheme also has a unique solution under the stated geometric assumption.Source: §8 · p. 84, §8.1.1 · p. 88
  • The fully discrete scheme produces equidistributed interface meshes in two dimensions and conformal polyhedral hypersurfaces in three dimensions.These mesh properties are established for the scheme (147).Source: §8.1.1 · p. 89

8.1.2 Semidiscrete finite element approximation

  • The semidiscrete interface remains conformal; in two dimensions, neighboring curve elements either have equal lengths or are parallel.The semidiscrete scheme also conserves the volume of the minus phase.Source: §8.1.2 · p. 92, §8.1.3 · p. 92

8.1.3 XFEMΓ for conservation of the phase volumes

  • XFEMΓ practically preserves both phase volumes and avoids spurious velocities at discrete stationary interfaces; constant-discrete-mean-curvature polyhedra, including bubble approximations, are stationary.For the fully discrete approximation, total mass conservation cannot be shown, although the two phase volumes are conserved well in practice.Source: §8.1.3 · pp. 92–94

8.1.4 Approximations based on the fluidic tangential velocity

  • An alternative fluidic-tangential-velocity formulation retains stability, but volume conservation requires an added condition and numerical integration on the interface.The formulation without that condition does not permit a proof of the stated volume-conservation result.Source: §8.1.4 · p. 94

8.2 Two-phase Navier–Stokes flow

  • The two-phase Navier–Stokes scheme has a unique solution, an unconditional stability bound when no bulk-mesh coarsening occurs, and equidistributed 2D or conformal 3D meshes.Its semidiscrete continuous-in-time version conserves both phase volumes exactly.Source: §8.2 · pp. 96–97

8.3 Alternative numerical approaches

  • Alternative two-phase methods include VOF, level-set, phase-field, and parametric interface tracking; the scheme also extends to soluble and insoluble surfactants.The cited examples include VOF by Hirt and Nichols (1981), level-set methods by Sussman et al. (1994), and surfactant cases by Barrett et al. (2015e, 2015d).Source: §8.3 · pp. 97–98

9 Willmore flow

This section develops finite element approximations for Willmore flow, including a stable weak formulation and a fully discrete scheme whose solvability and limitations are examined. It also treats spontaneous-curvature and area-difference effects, constrained and boundary cases, and notes level-set and phase-field methods as alternatives.

9.1 Derivation of the flow

  • Willmore flow is the L2-gradient flow of Willmore energy; in d = 2, it is often called elastic flow.Source: §9.1 · pp. 98–99

9.2 A finite element approximation of Willmore flow

  • The fully discrete approximation has a unique solution at each time step, while its semidiscrete variant satisfies Theorem 88(iii).The approximation uses a discrete Weingarten map based on one of the formulations in (35).Source: §9.2 · pp. 99–100

9.3 A stable approximation of Willmore flow

  • Dziuk’s (2008) stable semidiscrete approximation lacks good mesh properties because its motion is almost exclusively normal, which generally produces bad meshes.Source: §9.3 · pp. 100, 103
  • A new weak formulation is designed to yield semidiscretizations that are both stable and have good mesh properties; its scheme satisfies Theorem 88(iii).A sufficiently smooth solution of the formulation is also a solution of the strong Willmore-flow formulation.Source: §9.3 · pp. 103, 105, 107
  • The fully discrete Willmore-flow scheme has a unique solution at each time step, but no stability bound appears provable for it or its generalizations.The same stability limitation is noted for a fully discrete approximation of the Dziuk scheme.Source: §9.3 · p. 108

9.4 Willmore flow with spontaneous curvature and area difference elasticity effects

  • Spontaneous curvature generalizes the Willmore energy for membrane asymmetry, while ADE models penalize deviations from an optimal area difference, approximated by integrated mean curvature.The ADE model uses given constants β≥0 and M0; with β=0, the volume- and surface-area-preserving flow is called Helfrich flow.Source: §9.4 · pp. 109–110
  • The semidiscrete and fully discrete Dziuk-type schemes lack good mesh properties, motivating extensions of the mesh-quality approximation to include spontaneous curvature and ADE effects.The extended weak formulation and approximation can also incorporate volume and surface-area constraints while retaining stability.Source: §9.4 · pp. 115–116
  • For surfaces with boundary, fixed, Navier, clamped, and free conditions address boundary terms; the listed finite-element scheme adapts to Navier conditions, while the alternative approaches adapt to all three types.The latter approaches also have semidiscrete stability results.Source: §9.4 · pp. 116–117

9.5 Alternative numerical approaches

  • Rusu (2005) introduced a mixed variational Willmore formulation using position and mean-curvature vectors for continuous piecewise linear finite elements; later graph-based semidiscrete analyses assume a sufficiently smooth solution.Clarenz et al. (2004) extended the approach to surfaces with boundaries and restoration, while Deckelnick and Dziuk (2006), Deckelnick and Schieweck (2010), and Deckelnick et al. (2015) presented error analyses.Source: §9.5 · p. 117
  • Level-set methods were studied by Droske and Rumpf (2004), while phase-field approaches are discussed by Du et al. (2005), Esedo¯glu et al. (2014), and Bretin et al. (2015).Source: §9.5 · p. 118

10 Biomembranes

This section develops fluid-coupled models and finite-element formulations for biomembrane evolution, including the role of surface incompressibility in mesh behavior. It also addresses stability and implementation, two-phase extensions, and alternative approaches to vesicle and membrane evolution.

10.1 A model for the dynamics of fluidic biomembranes

  • The model couples bulk Stokes flow to tangential surface Stokes flow, surface incompressibility, and curvature forces; the two incompressibility conditions yield conservation without extra area or volume constraints.The model assumes the interface moves with the bulk velocity, whose normal component determines its evolution.Source: §10.1 · pp. 118, 119, 120, 122

10.2 A weak formulation for the dynamics of biomembranes

  • The natural weak formulation sets interface velocity V=u; surface incompressibility supports good meshes, while pressure uniqueness requires no spherical component of the membrane.Under that condition, surface pressure is unique and bulk pressure is unique up to an additive constant.Source: §10.2 · pp. 122, 124

10.3 Semidiscrete finite element approximation

  • The semidiscrete approximation has stability results; its virtual-element variant conserves surface area and enclosed volume and satisfies an energy identity.These conservation properties are stated for the virtual-element approach to the semidiscrete approximation.Source: §10.3 · pp. 126–127
  • Conservation of each basis-function support measure preserves equidistribution in 2D for odd J=K; a modified scheme conserves each element's measure for arbitrary J≥2 and d≥2.For the modified scheme, the trial space for the discrete pressure uses piecewise constants.Source: §10.3 · p. 127
  • Piecewise-linear surface velocity and pressure approximations may produce oscillatory pressure; quadratic bulk-velocity interpolation improves pressure behavior but can impair area conservation in d=3.Under a discrete LBB condition, the formulations have unique solutions; extensions include bulk Navier–Stokes flow, and Schur complements can solve the resulting linear systems.Source: §10.3 · pp. 127–128

10.4 Two-phase biomembranes

  • Two-phase biomembranes use a surface phase field with concentration-dependent energy parameters and Cahn–Hilliard dynamics for species concentration.The order parameter c takes values ±1 in the two phases, and the Ginzburg–Landau energy represents their interfacial energy.Source: §10.4 · pp. 128–129
  • Concentration-dependent membrane forces add tangential contributions that produce a Marangoni-type effect; a stable semidiscrete formulation and numerical computations show phase effects on membrane evolution.The generalized force is obtained from the energy variation with respect to the membrane.Source: §10.4 · pp. 129–130

10.5 Alternative numerical approaches

  • Alternative approaches address fluid-membrane coupling, local incompressibility, and surface or bulk viscosity using level sets, phase fields, immersed boundaries, or surface Stokes systems.The cited surface Stokes approaches enforce volume conservation with a global Lagrange multiplier; one study incorporating surface and bulk viscosity reports axisymmetric numerical results.Source: §10.5 · p. 130
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