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Many physical problems can be modelled by partial differential equations onunknown domains. Several examples can easily be found in the dynamics of freeinterfaces in fluid dynamics, solid mechanics or in fluid-solid interactions.To solve these equations in an arbitrary domain with nonlinear deformations, wepropose a mathematical approach allowing to track the boundary of the domain,analogue of, and complementary to, the Arbitrary Lagrangian-Eulerian (ALE)method for the interior of the domain. We name this method as the BoundaryArbitrary Lagrangian-Eulerian (BALE) method. Additionally, in many situationsnonlinear deformations can be avoided with the help of some analyses which relyon small deformations of the boundary, such as stability analysis, asymptoticexpansion and gradient-based shape optimisation. For these cases, we propose anapproach to perturb the domain and its boundaries and write the partialdifferential equations at the unperturbed domain together with the boundaryconditions at the unperturbed boundary, instead of at the perturbed ones, whichare a priori unknown. We name this method as the Deformable BoundaryPerturbation (DBP) method. These two proposed methods rely on the boundaryexterior differential operator, whose relevant properties for the present workare evidenced. We show an example for which the BALE and DBP methods areapplied, and for which we include the weak formulation revealing theappropriateness of the finite element method in this context.

PDEs on deformable domains: Boundary Arbitrary Lagrangian-Eulerian (BALE) and Deformable Boundary Perturbation (DBP) methods