In this paper we prove that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon-Miranda) maximum principle on convex polygonal domains. Using this result, we establish the stability of the Ritz projection with mixed boundary conditions in L∞ norm up to a logarithmic term.

We prove exponential rates of convergence of hp-dG interior penalty (IP) methods for second-order elliptic problems with mixed boundary conditions in polyhedra which are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic polynomial degree distributions of μ-bounded variation. Compared to homogeneous Dirichlet boundary conditions in [10, 11], for problems wi...

We investigate the “hot spots” property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the “hot spots” property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curv...

We consider the transient drift-diffusion model with fast diffusion terms. This problem is not only degenerate but also singular. We first present existence result for general nonlinear diffusivities for the Dirichlet-Neumann mixed boundary value problem. Then, the extinction phenomenon of weak solutions for the homogeneous Dirichlet boundary problem is studied. Sufficient conditions on the ext...

We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. When the interspecific competition parameter tends to infinity, the system solution converges to that of a freeboundary problem. If all stationary solutions of this limit problem are non-degenerate and if a certain linear combination of the boundary data do...

The Casimir effect giving rise to an attractive or repulsive force between the configuration boundaries that confine the massless scalar field is reexamined for one to three-dimensional pistons in this paper. Especially, we consider Casimir pistons with hybrid boundary conditions, where the boundary condition on the piston is Neumann and those on other surfaces are Dirichlet. We show that the C...

A Dirichlet-to-Neumann (DtN) condition is derived for the numerical solution of time-harmonic multiple scattering problems, where the scatterer consists of several disjoint components. It is obtained by combining contributions from multiple purely outgoing wave fields. The DtN condition yields an exact nonreflecting boundary condition for the situation, where the computational domain and its ex...

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...

In this article, computational results from boundary integral equations and their normal derivatives for the same test cases are compared. Both kinds of formulations are desingularized on their real boundary. The test cases are chosen as a uniform flow past a circular cylinder for both the Dirichlet and Neumann problems. The results indicate that the desingularized method for the standard bound...

We derive eigenvalue asymptotics for Sturm–Liouville operators with singular complex-valued potentials from the space W 2 (0, 1), α ∈ [0, 1], and Dirichlet or Neumann–Dirichlet boundary conditions. We also give application of the obtained results to the inverse spectral problem of recovering the potential by these two spectra.