PDE's on surfaces---a diffuse interface approach

Abstract

Abstract. We introduce a new approach to deal with the numerical solution of partial differential equations on surfaces. Thereby we reformulate the problem on a larger domain in one higher dimension and introduce a diffuse interface region of a phase-field variable, which is defined in the whole domain. The surface of interest is now only implicitly given by the 1/2-level set of this phase-field variable. Formal matched asymptotics show the convergence of the reformulated problem to the original PDE on the surface, as the diffuse interface width shrinks to zero. The main advantage of the approach is the possibility to formulate the problem on a Cartesian grid. With adaptive grid refinement the additional computational cost resulting from the higher dimension can be significantly reduced. Examples on linear diffusion and nonlinear phase separation demonstrate the wide applicability of the method.

Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology

“…Governing equations of the model are: (10) and the boundary condition for the variable c 2 is D 2 (∂c 2 / ∂N) | dΩ = R bind where N is the outward normal. No boundary conditions are needed for c 1 because dΩ is closed.…”

“…In simulations, V max = 1 s −1 , h = 3 μm and the masked region is shown in Fig 9. The rate of binding to the membrane, which enters the first equation in (10) and the boundary condition for c 2 , is R bind = −k off c 1 (x, t) + k on c 2 (x, t) where x ∈ dΩ.…”

“…For this, the first equation in (10) has been solved separately with R bind = 0 and D 1 = 0.25 μm 2 /s (in real experiments, a special molecular construct that does not dissociate from the membrane and has diffusive properties similar to those of Rac was used [17] ). The normalized fluorescence in the unbleached portion of the membrane dω (Fig 9) is where .…”

“…They also are suited well for problems involving moving membranes. Recently, the ideas of [7] have been further developed in [9] and applied in the context of phase field approximation in [10].…”

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

“…However, the method was only applicable to no-flux boundary conditions, and no further extensions to other types of equations or boundary conditions have been reported. Recently, Lowengrub and coworkers [27,28,29,30,31,32,33] developed an alternative formulation for solving partial differential equations with various boundary conditions, based on asymptotic analyses commonly conducted in phase field modeling, which is different from the general derivation of the smoothed boundary method presented in this paper. Although such an implementation for imposing boundary conditions differs from the 'formal' practice suggested by Cahn [17], it dramatically simplifies the formulation, provides a justification of the method, and increases the applicability of the approach.…”