The Laplace operator is considered for waveguides perturbed by a periodic structure consisting of N congruent obstacles spanning the waveguide. Neumann boundary conditions are imposed on the periodic structure, and either Neumann or Dirichlet conditions on the guide walls. It is proven that there are at least N (resp. N−1) trapped modes in the Neumann case (resp. Dirichlet case) under fairly ge...

We present a modular toolbox for parallel nite element simulations on distributed memory systems. The library named PadFEM includes a graphical editor for specifying domains with boundary conditions, automatic mesh generation, automaticmesh partitioning and mapping onto the processors of a MIMD-system. The parallel FEM-simulation uses the preconditioned conjugate gradient method. The paralleliz...

We consider the Laplacian in a curved two-dimensional strip of constant width squeezed between two curves, subject to Dirichlet boundary conditions on one of the curves and variable Robin boundary conditions on the other. We prove that, for certain types of Robin boundary conditions, the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Laplacian in a D...

We derive boundary conditions for the vorticity equation with solid wall boundaries. The formulation uses a Dirichlet condition for the normal component of vorticity, and Neumann type conditions for the tangential components. In a Galerkin (integral) formulation the tangential condition is natural, i.e. it is enforced by a right-hand side functional and does not impose a boundary constraint on ...

We consider a singular perturbed eigenvalue problem for Laplace operator in a cylinder with frequent interchange of type of boundary condition on a lateral surface. These boundary conditions are prescribed by partition of lateral surface in a great number of narrow strips on those the Dirichlet and Neumann conditions are imposed by turns. We study the case of the homogenized problem containing ...

We derive boundary conditions for the vorticity equation with solid wall boundaries. The formulation uses a Dirichlet condition for the normal component of vorticity, and Neumann type conditions for the tangential components. In a Galerkin (integral) formulation the tangential condition is natural, i.e. it is enforced by a right-hand side functional and does not impose a boundary constraint on ...

This work is aimed at the derivation of reliable and efficient a posteriori error estimates for convectiondominated diffusion problems motivated by a linear Fokker-Planck problem appearing in computational neuroscience. We obtain computable error bounds of functional type for the static and time-dependent case and for different boundary conditions (mixed and pure Neumann boundary conditions). F...

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to self-adjoint Laplacians −∆Θ,Ω in L (Ω; dx) with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains Ω ⊂ R, n ∈ N, n ≥ 2. Second, we extend Friedlander’s inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonl...

We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes espe...

In this article we study the exact controllability of a one-dimensional wave equation with mixed boundary conditions in a non-cylindrical domain. The fixed endpoint has a Dirichlet-type boundary condition, while the moving end has a Neumann-type condition. When the speed of the moving endpoint is less than the characteristic speed, the exact controllability of this equation is established by Hi...