Maria Rosaria Lancia

A Transmission problem with a fractal interface

In this paper we study a transmission problem with a fractal interface K, where a second order transmission condition is imposed. We consider the case in which the interface K is the Koch curve and we prove existence and uniqueness of the weak solution of the problem in V (Ω, K), a suitable ”energy space”. The link between the variational formulation and the problem is possible once we recover a version of the Gauss-Green formula for fractal boundaries, hence a definition of ”normal derivative”.

Irregular Heat Flow Problems

We study a nonsteady transmission problem across either a fractal layer S or the corresponding prefractal layer

Numerical approximation of transmission problems across Koch-type highly conductive layers

S_h

An optimal mesh generation algorithm for domains with Koch type boundaries

. The transmission condition is of order two. Existence, uniqueness, and regularity results for the strict solution, in both cases, are established as well as convergence results for the solutions of the approximating problems in varying Hilbert spaces.

On the Removal of the Trailing Edge Singularity in 3D Flows

Abstract We prove a priori error estimates for a parabolic second order transmission problem across a prefractal interface Kn of Koch type which divides a given domain Ω into two non-convex sub-domains Ω n i . By exploiting some regularity results for the solution in Ω n i we build a suitable mesh, compliant with the so-called “Grisvard” conditions, which allows to achieve an optimal rate of convergence for the semidiscrete approximation of the prefractal problem by Galerkin method. The discretization in time is carried out by the θ-method.

Gleanings of radial cavitation

In this paper we propose a mesh algorithm to generate a regular and conformal family of nested triangulations for a planar domain divided into two non-convex polygonal subdomains by a prefractal Koch type interface. The presence of the interface, a polygonal curve, induces a natural triangulation in which the vertices of the prefractal are also nodes of the triangulation. In order to achieve an optimal rate of convergence of the numerical approximation a suitably refined mesh around the reentrant corners is required. This is achieved by generating a mesh compliant with the Grisvards condition. We present the mesh algorithm and a detailed proof of the Grisvard conditions.

Semilinear Evolution Problems with Ventcel-Type Conditions on Fractal Boundaries

The steady incompressible inviscid flow past a 3D airfoil with a sharp trailing edge TE is not uniquely determined by the free stream velocity U, unless some information about the shed vorticity is added. Namely, the concentrated vorticity ω normal to TE, which forms the vortex sheet released from the airfoil in steady state conditions, is an extra unknown to be determined by the solution. In fact, the irrotational flow is unique, contrary to what happens in 2D, but the real flow including the wake is not, and a Kutta condition is needed in order to determine ω and to retrieve uniqueness.

Fractal snowflake domain diffusion with boundary and interior drifts

It is shown that the constitutive character inducing radial cavitation of elastic balls is carried by the liquid-like portion, if any, of their stored-energy density.

Energy Forms on Non Self-similar Fractals

A semilinear parabolic transmission problem with Ventcels boundary conditions on a fractal interface or the corresponding prefractal interface is studied. Regularity results for the solution in both cases are proved. The asymptotic behaviour of the solutions of the approximating problems to the solution of limit fractal problem is analyzed.

Numerical Approximation of Boundary Integral Equations in Three Dimensional Aerodynamics

Abstract We study a parabolic Ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain. We prove that under standard conditions a related Cauchy problem possesses a unique classical solution and explain in which sense it solves a rigorous formulation of the initial Ventsell problem. As a second result we prove that functions that are intrinsically Lipschitz on the snowflake boundary admit Euclidean Lipschitz extensions to the closure of the entire domain. Our methods combine the fractal membrane analysis, the vector analysis for local Dirichlet forms and PDE on fractals, coercive closed forms, and the analysis of Lipschitz functions.