We prove an explicit formula for the spectral expansions in L(R) generated by selfadjoint differential operators (−1) d dx2n + n−1

We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The adaptivity is driven by the residual error estimator. We prove convergence even with optimal algebraic convergence rates. In particular, our analysis covers gene...

The transmission eigenvalue problem is a type of non-elliptic and non-selfadjoint spectral that arises in the wave scattering theory when invisibility/transparency occurs. eigenfunctions are interior resonant modes inside medium. We concerned with geometric rigidity show they concentrate on boundary surface underlying domain two senses. This substantiates recent numerical discovery Chow et al. ...

In this paper we consider the spectral properties of a class of non-local uniformly elliptic operators, which arise from the study of non-local uniformly elliptic partial differential equations. Such equations arise naturally in the study of a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded pertur...

In this work we analyse a method to construct a numerically efficient and computationally cheap sparse approximations of some of the matrix blocks arising in the block-factorised preconditioners for matrices with a two-by-two block structure. The matrices arise from finite element discretizations of partial differential equations. We consider scalar elliptic problems, however the approach is ap...

We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational...

In this work we prove that the eigenvalues of $n$-dimensional massive Dirac operator $\mathscr{D}_0 + V$, $n\ge2$, perturbed by a possibly non-Hermitian potential $V$, are localized in union two disjoint disks complex plane, provided $V$ is sufficiently small with respect to mixed norms $L^1_{x_j} L^\infty_{\widehat{x}_j}$, for $j\in\{1,\dots,n\}$. massless case, instead discrete spectrum empty...