Stefano Giani

A Convergent Adaptive Method for Elliptic Eigenvalue Problems

We prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second-order symmetric elliptic partial differential operators. The weak form is assumed to yield a bilinear form which is bounded and coercive in

hp-Version composite discontinuous Galerkin methods for elliptic problems on complicated domains.

H^1

Energy norm a posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions.

. Each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation. The error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems, and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem. Because of this nonlinearity, the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine.

Domain Decomposition Preconditioners for Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

In this paper we introduce the

Adaptive finite element methods for computing band gaps in photonic crystals

hp

An a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems

-version discontinuous Galerkin composite finite element method for the discretization of second-order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. While standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. The key idea in the construction of this latter class of methods is that the computational domain

Boundary element dynamical energy analysis: A versatile method for solving two or three dimensional wave problems in the high frequency limit

\Omega

Dynamical energy analysis for built-up acoustic systems at high frequencies

is no longer resolved by the mesh; instead, the finite element basis (or shape) functions are adapted to the geometric details present in

Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains

\Omega

An a-posteriori error estimate for hp-adaptive DG methods for convection-diffusion problems on anisotropically refined meshes

. In this paper, we extend these ideas to the discontinuous Galer...