Abstract In this paper, we compute the band structure of one- and two-dimensional phononic composites using extended finite element method (X-FEM) on structured higher-order (spectral) meshes. On partition-of-unity enrichment in analysis, X-FEM permits use meshes that do not conform to geometry holes inclusions. This eliminates need for remeshing shape optimization topology studies. two dimensi...

We study the dispersion and dissipation of the numerical scheme obtained by taking a weighted averaging of the consistent (finite element) mass matrix and lumped (spectral element) mass matrix for the small wavenumber limit. We find and prove that for the optimum blending the resulting scheme (a) provides 2p+4 order accuracy for p-th order method (two orders more accurate compared with finite a...

Partial Differential Equations and the Finite Element Method provides a much–needed, clear, and systematic introduction to modern theory of partial differential equations (PDEs) and finite element methods (FEM). Both nodal and hierachic concepts of the FEM are examined. Reflecting the growing complexity and multiscale nature of current engineering and scientific problems, the author emphasizes ...

Spectral clustering methods use eigenvectors of a matrix, called Gaussian affinity matrix, in order to define a low-dimensional space in which data points can be clustered. This matrix is widely used and depends on a free parameter σ. It is usually interpreted as some discretization of the Heat Equation Green kernel. Combining tools from Partial Differential Equations and Finite Elements theory...

We present constructive a priori error estimates forH 0 -projection into a space of polynomials on a one-dimensional interval. Here, “constructive” indicates that we can obtain the error bounds in which all constants are explicitly given or are represented in a numerically computable form. Using the properties of Legendre polynomials, we consider a method by which to determine these constants t...

We present a simple new approach to the solution of a wide class of spectral and resonance problems on infinite domains with regular ends, including those found in the study of quantum switches, waveguides, and acoustic scatterers. Our algorithm is part analytical and part numerical and is essentially a combination of four classical approaches (domain decomposition, boundary elements, finite el...