Section: New Results

Numerical analysis and simulation of heterogeneous systems

Participants : Xavier Antoine, Mohamed El Bouajaji, Karim Ramdani, Qinglin Tang, Julie Valein, Chi-Ting Wu.

In optics, metamaterials (also known as negative or left-handed materials), have known a growing interest in the last two decades. These artificial composite materials exhibit the property of having negative dielectric permittivity and magnetic permeability in a certain range of frequency, leading hence to materials with negative refractive index and super lens effects. In [8], Bunoiu (IECL, Metz, France) and Ramdani consider a complex wave system involving such materials. More precisely, they consider a periodic homogenization problem involving two isotropic materials with conductivities of different signs: a classical material and a metamaterial (or negative material). Combining the 𝐓-coercivity approach and the unfolding method for homogenization, they prove well-posedness results for the initial and the homogenized problems and obtain a convergence result, provided that the contrast between the two conductivities is large enough (in modulus).

In [18], Tucsnak (Institut de Mathématiques de Bordeaux, France), Valein and Wu study the numerical approximation of the solutions of a class of abstract parabolic time-optimal control problems with unbounded control operator. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the optimal time and the optimal controls of the approximate time optimal problems converge (in appropriate norms) to the optimal time and to the optimal controls of the original problem. In order to prove our main theorem, we provide a nonsmooth data error estimate for abstract parabolic systems.

In [4], Antoine and Lorin (School of Mathematics and Statistics, Ottawa, and CRM, Montréal, Canada) analyze the convergence of optimized Schwarz domain decomposition methods for the simulation of the time-domain Schrödinger equation with high-order local transmission conditions.

In [5], Antoine, Tang and Zhang (WPI, Austria and IRMAR, France) develop some spectral methods for computing the ground states and dynamics of space fractional Gross-Pitaevskii equations arising in the modeling of fractional Bose-Einstein equations with long-range nonlinear interactions. In addition, we also state some existence and uniqueness properties for the ground states of such equations, and prove some dynamical laws.

In [6], Bao (Department of Mathematics, Singapore), Tang and Zhang (WPI, Austria and IRMAR, France) develop a new efficient and spectrally accurate numerical for computing the ground state and dynamics of dipolar Bose-Einstein condensates. They pay a particular attention to the computation of the nonlinear nonlocal interactions through the use of the nonuniform fast Fourier transform.

In [22], Antoine, Levitt (CERMICS, France) and Tang derive a highly accurate and efficient new numerical method for computing the ground states of the fast rotating Gross-Pitaevskii equation. The method is based on a preconditioned nonlinear conjugate gradient method which leads to a high gain compared to most recent approaches.

In [26], Bao (Department of Mathematics, Singapore), Cai (Department of mathematics Purdue University, USA and CSRC, Beijing, China), Jia (Department of Mathematics, Singapore), Tang develop a uniformly accurate multiscale time integrator in conjunction with a spectral method for computing the dynamics of the nonrelativistic Dirac equation. The same authors develop and compare, in [27], some new numerical methods for the simulation of the Dirac equation when the nonrelativistic regime is considered.

The article [17] is devoted to explain how the open finite element solver GetDDM works. The mathematical methods behind GetDDM are optimized Schwarz domain decomposition methods with well-designed transmission boundary conditions. GetDDM allows to solve large scale high frequency wave problems (e.g. acoustics, electromagnetism, elasticity problems) on large clusters. This papers explains through examples and scripts how GetDDM must be used. GetDDM is a result of a long term collaboration between Xavier Antoine and Christophe Geuzaine (University of Liège).