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In this article we consider regularizations of the Dirac delta distribution with applications to prototypical elliptic and hyperbolic partial differential equations (PDEs). We study the convergence of a sequence of distributions S H to a singular term S as a parameter H (associated with the support size of S H ) shrinks to zero. We characterize this convergence in both the weak-⁎ topology of distributions and a weighted Sobolev norm. These notions motivate a framework for constructing regularizations of the delta distribution that includes a large class of existing methods in the literature. This framework allows different regularizations to be compared. The convergence of solutions of PDEs with these regularized source terms is then studied in various topologies such as pointwise convergence on a deleted neighborhood and weighted Sobolev norms. We also examine the lack of symmetry in tensor product regularizations and effects of dissipative error in hyperbolic problems. [ABSTRACT FROM AUTHOR]

Abstract: White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical systems in space dimensions d =1, 2, 3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d ⩾2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen–Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that shrinking the mesh size in simulations of the two-dimensional white noise-driven Allen–Cahn equation does not lead to the recovery of a physically meaningful limit. [Copyright &y& Elsevier]

Abstract: We present a new algorithm for the evaluation of the quasi-periodic Green function for a linear array of acoustic point sources such as those arising in the analysis of line array loudspeakers. A variety of classical algorithms (based on spatial and spectral representations, Ewald transformation, etc.) have been implemented in the past to evaluate these acoustic fields. However as we show, these methods become unstable and/or impractically expensive as the frequency of use of the sources increases. Here we introduce a new numerical scheme that overcomes some of these limitations allowing for simulations at unprecedentedly large frequencies. The method is based on a new integral representation derived from the classic spatial form, and on suitable further manipulations of the relevant integrands to render the integrals amenable to efficient and accurate approximations through standard quadrature formulas. We include a variety of numerical results that demonstrate that our algorithm compares favorably with several classical methods both for points close to the line where the poles are located and at high-frequencies while remaining competitive with them in every other instance. [Copyright &y& Elsevier]

The authors present a new method for the realization of exact non-reflecting (transparent) boundary conditions in two dimensional direct scattering problems. This work is an extension of Keller, Givoli, and Grote’s work on such conditions which required that the shape of the boundary be quite specific, i.e. circular or elliptical. The condition is enforced via the Dirichlet–Neumann operator (DNO) which, on general boundaries, presents the main difficulty in the method. The implementation is performed by one of two perturbative methods (where the perturbation parameter measures the deformation of the general geometry from a canonical one). A rigorous proof of the analyticity for the DNO with respect to this perturbation parameter is presented. Numerical results show both perturbative methods are fast and accurate, and can enable significant computational savings. [Copyright &y& Elsevier]

We present a novel integral-equation algorithm for evaluation of Zaremba eigenvalues and eigenfunctions , that is, eigenvalues and eigenfunctions of the Laplace operator with mixed Dirichlet–Neumann boundary conditions; of course, (slight modifications of) our algorithms are also applicable to the pure Dirichlet and Neumann eigenproblems. Expressing the eigenfunctions by means of an ansatz based on the single layer boundary operator, the Zaremba eigenproblem is transformed into a nonlinear equation for the eigenvalue μ . For smooth domains the singular structure at Dirichlet–Neumann junctions is incorporated as part of our corresponding numerical algorithm—which otherwise relies on use of the cosine change of variables, trigonometric polynomials and, to avoid the Gibbs phenomenon that would arise from the solution singularities, the Fourier Continuation method (FC). The resulting numerical algorithm converges with high order accuracy without recourse to use of meshes finer than those resulting from the cosine transformation. For non-smooth (Lipschitz) domains, in turn, an alternative algorithm is presented which achieves high-order accuracy on the basis of graded meshes. In either case, smooth or Lipschitz boundary, eigenvalues are evaluated by searching for zero minimal singular values of a suitably stabilized discrete version of the single layer operator mentioned above. (The stabilization technique is used to enable robust non-local zero searches.) The resulting methods, which are fast and highly accurate for high- and low-frequencies alike, can solve extremely challenging two-dimensional Dirichlet, Neumann and Zaremba eigenproblems with high accuracies in short computing times—enabling, in particular, evaluation of thousands of eigenvalues and corresponding eigenfunctions for a given smooth or non-smooth geometry with nearly full double-precision accuracy. [ABSTRACT FROM AUTHOR]

• To determine the right mixed boundary conditions on the surface of a container, so that any shock inside yields a resonance. • To develop an efficient algorithm for optimizing the Green's function for a mixed Steklov-Neumann problem. • To derive an original asymptotic formula for the Steklov-Neumann function. This paper examines the Laplace equation with mixed boundary conditions, the Neumann and Steklov boundary conditions. This models a container with holes in it, like a pond filled with water but partly covered by immovable pieces on the surface. The main objective is to determine the right extent of the covering pieces, so that any shock inside the container yields a resonance. To this end, an algorithm is developed which uses asymptotic formulas concerning perturbations of the partitioning of the boundary pieces. Proofs for these formulas are established. Furthermore, this paper displays some results concerning bounds and examples with regards to the governing problem. [ABSTRACT FROM AUTHOR]