Trefftz methods are finite element-type schemes whose test and trial functions are (locally)
solutions of the targeted differential equation. They are particularly popular for time …

There has been much recent research on preconditioning discretisations of the Helmholtz
operator Δ+ k^ 2 Δ+ k 2 (subject to suitable boundary conditions) using a discrete version of …

The phenomenon of irregular frequencies or spurious modes when solving the Kirchhoff–
Helmholtz integral equation has been extensively studied over the last six or seven …

We consider physics-informed neural networks (PINNs)(Raissiet al., 2019) for forward
physical problems. In order to find optimal PINNs configuration, we introduce a hyper …

We discuss the stability theory and numerical analysis of the Helmholtz equation with
variable and possibly nonsmooth or oscillatory coefficients. Using the unique continuation …

We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, ie the
equation∇⋅(A∇ u)+ k 2 nu=− f where both A and n are functions of position. We prove new …

We consider the Helmholtz transmission problem with one penetrable star-shaped Lipschitz
obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds …

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in
the sense that the bilinear forms cannot be bounded below by a positive multiple of the …

6.1▪“When All Else Fails, Integrate by Parts”: New and Old Variational Formulations for
Linear Elliptic PDEs 6.1. 1▪ Introduction Although the motto “when all else fails, integrate by …

We consider three problems for the Helmholtz equation in interior and exterior domains in
R^d (d=2,3): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for …