Parametric Finite Element Approximation Research Articles

Parametric finite element approximations for anisotropic surface diffusion with axisymmetric geometry

We consider parametric finite element approximations for the anisotropic surface diffusion flow in an axisymmetric setting. Based on the anisotropy function, we introduce a symmetric positive definite matrix with a suitable stabilizing function. This then gives rise to two novel weak formulations for the axisymmetric flow in terms of the generating curve of the interface. By using piecewise linear elements in space and backward Euler in time, we discretize the weak formulations to obtain different approximating methods. These include a linear approximation which has good mesh properties and nonlinear approximations which can be shown rigorously to satisfy the volume preservation or the unconditional energy stability on the discrete level. Extensive numerical results are reported to demonstrate the accuracy and efficiency of the introduced methods for computing the axisymmetric flow.

A New Approach to the Analysis of Parametric Finite Element Approximations to Mean Curvature Flow

A New Approach to the Analysis of Parametric Finite Element Approximations to Mean Curvature Flow

Structure-preserving discretizations of two-phase Navier–Stokes flow using fitted and unfitted approaches

We consider the numerical approximation of a sharp-interface model for two-phase flow, which is given by the incompressible Navier–Stokes equations in the bulk domain together with the classical interface conditions on the interface. We propose structure-preserving finite element methods for the model, meaning in particular that volume preservation and energy decay are satisfied on the discrete level. For the evolving fluid interface, we employ parametric finite element approximations that introduce an implicit tangential velocity to improve the quality of the interface mesh. For the two-phase Navier–Stokes equations, we consider two different approaches: an unfitted and a fitted finite element method, respectively. In the unfitted approach, the constructed method is based on an Eulerian weak formulation, while in the fitted approach a novel arbitrary Lagrangian–Eulerian (ALE) weak formulation is introduced. Using suitable discretizations of these two formulations, we introduce two finite element methods and prove their structure-preserving properties. Numerical results are presented to show the accuracy and efficiency of the introduced methods.

A symmetrized parametric finite element method for simulating solid-state dewetting problems

We propose an accurate, area/mass-conservative and energy-dissipative parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting with arbitrary anisotropic surface energy. The model is governed by the anisotropic surface diffusion equation with suitable boundary conditions at the contact points, and could be used to describe the kinetic evolution of the film/vapor interface with contact line migration. The sharp-interface continuum model is firstly reformulated into a conservative form by introducing a symmetric positive definite matrix Zk(n) that depends on the Cahn–Hoffman ξ-vector and a stabilizing function k(n). A new symmetrized variational formulation with arbitrary anisotropic surface energy is built and then its area/mass conservation and energy dissipation properties are proved. By using piecewise linear elements in space, we propose the spatial semi-discretized scheme, and using backward Euler in time with an appropriate approximation of the unit normal vector, we further derive the fully discretized parametric finite element approximation. We show, theoretically, that the semi-discretized and fully discretized schemes are both area/mass-conservative and energy-dissipative. Finally, numerical experiments are reported to show the accuracy and efficiency of the numerical method as well as the anisotropic effects on the morphological evolution processes for solid-state dewetting of thin films.

Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation

A finite element method for the evolution of a two-phase membrane in a sharp interface formulation is introduced. The evolution equations are given as an $L^2$--gradient flow of an energy involving an elastic bending energy and a line energy. In the two phases Helfrich-type evolution equations are prescribed, and on the interface, an evolving curve on an evolving surface, highly nonlinear boundary conditions have to hold. Here we consider both $C^0$-- and $C^1$--matching conditions for the surface at the interface. A new weak formulation is introduced, allowing for a stable semidiscrete parametric finite element approximation of the governing equations. In addition, we show existence and uniqueness for a fully discrete version of the scheme. Numerical simulations demonstrate that the approach can deal with a multitude of geometries. In particular, the paper shows the first computations based on a sharp interface description, which are not restricted to the axisymmetric case.

Finite element approximation for the dynamics of asymmetric fluidic biomembranes

We present a parametric finite element approximation of a fluidic membrane, whose evolution is governed by a surface Navier–Stokes equation coupled to bulk Navier–Stokes equations. The elastic properties of the membrane are modelled with the help of curvature energies of Willmore and Helfrich type. Forces stemming from these energies act on the surface fluid, together with a forcing from the bulk fluid. Using ideas from PDE constrained optimization, a weak formulation is derived, which allows for a stable semi-discretization. An important new feature of the present work is that we are able to also deal with spontaneous curvature and an area-difference elasticity contribution in the curvature energy. This allows for the modelling of asymmetric membranes, which compared to the symmetric case lead to quite different shapes. This is demonstrated in the numerical computations presented.

Stable finite element approximations of two-phase flow with soluble surfactant

A parametric finite element approximation of incompressible two-phase flow with soluble surfactants is presented. The Navier–Stokes equations are coupled to bulk and surfaces PDEs for the surfactant concentrations. At the interface adsorption, desorption and stress balances involving curvature effects and Marangoni forces have to be considered. A parametric finite element approximation for the advection of the interface, which maintains good mesh properties, is coupled to the evolving surface finite element method, which is used to discretize the surface PDE for the interface surfactant concentration. The resulting system is solved together with standard finite element approximations of the Navier–Stokes equations and of the bulk parabolic PDE for the surfactant concentration. Semidiscrete and fully discrete approximations are analyzed with respect to stability, conservation and existence/uniqueness issues. The approach is validated for simple test cases and for complex scenarios, including colliding drops in a shear flow, which are computed in two and three space dimensions.

On the stable numerical approximation of two-phase flow with insoluble surfactant

We present a parametric finite element approximation of two-phase flow with insoluble surfactant. This free boundary problem is given by the Navier--Stokes equations for the two-phase flow in the bulk, which are coupled to the transport equation for the insoluble surfactant on the interface that separates the two phases. We combine the evolving surface finite element method with an approach previously introduced by the authors for two-phase Navier--Stokes flow, which maintains good mesh properties. The derived finite element approximation of two-phase flow with insoluble surfactant can be shown to be stable. Several numerical simulations demonstrate the practicality of our numerical method.

A Stable Parametric Finite Element Discretization of Two-Phase Navier–Stokes Flow

We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Navier--Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. The parametric finite element approximation of the evolving interface is then coupled to a standard finite element approximation of the two-phase Navier--Stokes equations in the bulk. Here enriching the pressure approximation space with the help of an XFEM function ensures good volume conservation properties for the two phase regions. In addition, the mesh quality of the parametric approximation of the interface in general does not deteriorate over time, and an equidistribution property can be shown for a semidiscrete continuous-in-time variant of our scheme in two space dimensions. Moreover, our finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results in two and three space dimensions.

Eliminating spurious velocities with a stable approximation of viscous incompressible two-phase Stokes flow

We present a parametric finite element approximation of two-phase flow. This free boundary problem is given by the Stokes equations in the two phases, which are coupled via jump conditions across the interface. Using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface. In addition, the mesh quality of the parametric approximation of the interface does not deteriorate, in general, over time; and an equidistribution property can be shown for a semidiscrete continuous-in-time variant of our scheme in two space dimensions. Moreover, on using a simple XFEM pressure space enrichment, we obtain exact volume conservation for the two phase regions. Furthermore, our fully discrete finite element approximation can be shown to be unconditionally stable. We demonstrate the applicability of our method with some numerical results which, in particular, demonstrate that spurious velocities can be avoided in the classical test cases.

Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves

Deckelnick and Dziuk (Math. Comput. 78(266):645–671, 2009) proved a stability bound for a continuous-in-time semidiscrete parametric finite element approximation of the elastic flow of closed curves in $${\mathbb{R}^d, d\geq2}$$ . We extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in Barrett et al. (J Comput Phys 222(1): 441–462, 2007; SIAM J Sci Comput 31(1):225–253, 2008; IMA J Numer Anal 30(1):4–60, 2010), in particular an equidistribution mesh property. For this new approximation, we obtain also a stability bound for a continuous-in-time semidiscrete scheme. Apart from the isotropic situation, we also consider the case of an anisotropic elastic energy. In addition to the evolution of closed curves, we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions.

The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute

AbstractOn the basis of our previous work, we introduce novel fully discrete, fully practical parametric finite element approximations for geometric evolution equations of curves in the plane. The fully implicit approximations are unconditionally stable and intrinsically equidistribute the vertices at each time level. We present iterative solution methods for the systems of nonlinear equations arising at each time level and present several numerical results. The ideas easily generalize to the evolution of curve networks and to anisotropic surface energies. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies

We present a variational formulation for the evolution of surface clusters in R3 by mean curvature flow, surface diffusion and their anisotropic variants. We introduce the triple junction line conditions that are induced by the considered gradient flows, and present weak formulations of these flows. In addition, we consider the case where a subset of the boundaries of these clusters are constrained to lie on an external boundary. These formulations lead to unconditionally stable, fully discrete, parametric finite element approximations. The resulting schemes have very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments, including isotropic double, triple and quadruple bubbles, as well as clusters evolving under anisotropic mean curvature flow and anisotropic surface diffusion, including for regularized crystalline surface energy densities.

On stable parametric finite element methods for the Stefan problem and the Mullins–Sekerka problem with applications to dendritic growth

We introduce a parametric finite element approximation for the Stefan problem with the Gibbs–Thomson law and kinetic undercooling, which mimics the underlying energy structure of the problem. The proposed method is also applicable to certain quasi-stationary variants, such as the Mullins–Sekerka problem. In addition, fully anisotropic energies are easily handled. The approximation has good mesh properties, leading to a well-conditioned discretization, even in three space dimensions. Several numerical computations, including for dendritic growth and for snow crystal growth, are presented.

Numerical approximation of gradient flows for closed curves in Rd

We present parametric finite element approximations of curvature flows for curves in Rd, d ≥ 2, as well as for curves on two-dimensional manifolds in R3. Here we consider the curve shortening flow, curve diffusion and the elastic flow. It is demonstrated that the curve shortening and the elastic flows on manifolds can be used to compute nontrivial geodesics, and that the corresponding geodesic curve diffusion flow leads to solutions of partitioning problems on two-dimensional manifolds in R3. In addition, we extend these schemes to anisotropic surface energy densities. The presented schemes have very good properties with respect to stability and the distribution of mesh points, and hence no remeshing is needed in practice.

Mini-Workshop: Mathematics of Biological Membranes

The focus of this workshop was the modeling, analysis and nu- merical simulation of biological membranes. The fundamental models which characterize equilibria are based on bending energies such as the Helfrich energy and suitable generalizations. Talks in the workshop focused on equi- librium shapes of membranes and vesicles, the corresponding relaxation dy- namics and the coupling with outer or inner flows.

Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces

We define higher-order analogues to the piecewise linear surface finite element method studied in [G. Dziuk, “Finite elements for the Beltrami operator on arbitrary surfaces,” in Partial Differential Equations and Calculus of Variations, Springer-Verlag, Berlin, 1988, pp. 142-155] and prove error estimates in both pointwise and $L_2$-based norms. Using the Laplace-Beltrami problem on an implicitly defined surface $\Gamma$ as a model PDE, we define Lagrange finite element methods of arbitrary degree on polynomial approximations to $\Gamma$ which likewise are of arbitrary degree. Then we prove a priori error estimates in the $L_2$, $H^1$, and corresponding pointwise norms that demonstrate the interaction between the “PDE error” that arises from employing a finite-dimensional finite element space and the “geometric error” that results from approximating $\Gamma$. We also consider parametric finite element approximations that are defined on $\Gamma$ and thus induce no geometric error. Computational examples confirm the sharpness of our error estimates.

A variational formulation of anisotropic geometric evolution equations in higher dimensions

We present a novel variational formulation of fully anisotropic motion by surface diffusion and mean curvature flow in $$\mathbb {R}^d$$, d ≥ 2. This new formulation leads to an unconditionally stable, fully discrete, parametric finite element approximation in the case d = 2 or 3. The resulting scheme has very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments for d = 3, including regularized crystalline mean curvature flow and regularized crystalline surface diffusion.

On the parametric finite element approximation of evolving hypersurfaces in [formula omitted

We present a variational formulation of motion by minus the Laplacian of curvature and mean curvature flow, as well as related second and fourth order flows of a closed hypersurface in R3. On introducing a parametric finite element approximation, we prove stability bounds and compare our scheme with existing approaches. The presented scheme has very good properties with respect to the distribution of mesh points and, if applicable, volume conservation.

On the Variational Approximation of Combined Second and Fourth Order Geometric Evolution Equations

We present a variational formulation of combined motion by minus the Laplacian of curvature and mean curvature flow, as well as related flows. The proposed scheme covers both the closed curve case and the case of curves that are connected via triple or quadruple junction points or intersect the external boundary. On introducing a parametric finite element approximation, we prove stability bounds and compare our scheme with existing approaches. The presented scheme has very good properties with respect to the equidistribution of mesh points and, if applicable, area conservation.