UC Merced

Understanding light–matter interactions in plasmonic nanostructures and metamaterials is crucial for a wide range of applications from biochemical sensing and high resolution imaging to optical cloaking. This leads us to study Wave scattering by negative materials, which in turn leads to a Helmholtz equation with sign-changing coefficients, a regime in which classical coercivity of the variational formulation breaks down at resonance frequencies. This thesis develops and analyzes Galerkin methods to address the theoretical and computational challenges of this near-resonant plasmonic scattering problem. On the theoretical side, we characterize the near-resonance behavior by constructing the resonances and associated resonant modes for canonical geometries (e.g. a penetrable ball). As the material parameter ϵm approaches a critical value, the solution exhibits unbounded growth in its amplitude, with a loss of coercivity and can admit, for some wavenumbers, a plasmonic resonance. To give us well-posedness in the near region surrounding these pathological points, we re-introduce the technique of T-coercivity, developed in \cite{BoChCi12,BCCC16,BCC18} for this problem, where we may show we have Fredholm solvability. These theoretical results (including an adaption of the T-coercivity theory to the discrete setting) lay the groundwork for developing robust numerical schemes is the plasmonic regime.On the computational side, we design and investigate several Galerkin strategies for this sign-changing Helmholtz problem, including a standard FEM method, an enriched FEM (in 3D) and a Trefftz DG method (in 2D). The standard $H^1$ FEM is used as a baseline, but fails in the near-resonance case. In this near-resonant regime, the solution develops rapid oscillations and strongly localized fields (surface plasmons) that are difficult to capture with a low-order polynomial basis, often requiring prohibitively fine meshes. In 3D, We propose an enriched FEM, where we augment the usual FEM polynomial space with the analytical surface plasmon to encapsulate the dominant scattering mod. We derive a modified weak formulation in this extended function space and show that the resulting enriched system couples the standard H1 solution with the added plasmonic basis function.In parallel in 2D, we develop a Trefftz-DG method as an alternative Galerkin approach tailored to the Helmholtz operator. In this method, we abandon the polynomial basis and instead use Trefftz basis functions – specifically, propagative and evanescent plane waves – that exactly satisfy the local Helmholtz equation inside each element. This plane-wave enriched DG formulation more naturally embeds the oscillatory and decaying components of the solution into the approximation space. We extend the notion of T-coercivity to the discontinuous Galerkin setting to handle the sign-changing coefficient, and we prove a number of fundamental results for the Trefftz DG problem in the sign-changing case, leading to both a quasi-optimal convergence estimate for the Trefftz-DG scheme under an augmented norm (demonstrating stability of the method in the T-coercive case) and control of the L2 norm by the method. Finally, we present numerical results for the Trefftz PWDG method, both for the sign-changing method in general, and for the near-resonance case in particular.