Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation

We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a Lax-Richtmyer type equivalence theorem -- a consistent method is convergent if and only if it is stable. We define consistency as convergence on a dense subspace and stability as discrete well-posedness. In some applications convergence is harder to prove than consistency or stability since convergence requires knowledge of the solution. An equivalence theorem can be useful in such settings. We give concrete instances of equivalence theorems for polynomial interpolation, numerical differentiation, numerical integration using quadrature rules and Monte Carlo integration.Comment: 18 page

PyDEC: Software and Algorithms for Discretization of Exterior Calculus

This paper describes the algorithms, features and implementation of PyDEC, a Python library for computations related to the discretization of exterior calculus. PyDEC facilitates inquiry into both physical problems on manifolds as well as purely topological problems on abstract complexes. We describe efficient algorithms for constructing the operators and objects that arise in discrete exterior calculus, lowest order finite element exterior calculus and in related topological problems. Our algorithms are formulated in terms of high-level matrix operations which extend to arbitrary dimension. As a result, our implementations map well to the facilities of numerical libraries such as NumPy and SciPy. The availability of such libraries makes Python suitable for prototyping numerical methods. We demonstrate how PyDEC is used to solve physical and topological problems through several concise examples.Comment: Revised as per referee reports. Added information on scalability, removed redundant text, emphasized the role of matrix based algorithms, shortened length of pape

Numerical Experiments for Darcy Flow on a Surface Using Mixed Exterior Calculus Methods

There are very few results on mixed finite element methods on surfaces. A theory for the study of such methods was given recently by Holst and Stern, using a variational crimes framework in the context of finite element exterior calculus. However, we are not aware of any numerical experiments where mixed finite elements derived from discretizations of exterior calculus are used for a surface domain. This short note shows results of our preliminary experiments using mixed methods for Darcy flow (hence scalar Poisson's equation in mixed form) on surfaces. We demonstrate two numerical methods. One is derived from the primal-dual Discrete Exterior Calculus and the other from lowest order finite element exterior calculus. The programming was done in the language Python, using the PyDEC package which makes the code very short and easy to read. The qualitative convergence studies seem to be promising.Comment: 14 pages, 11 figure

Electron-Ion Recombination Rate Coefficients and Photoionization Cross Sections for Astrophysically Abundant Elements. VII. Relativistic calculations for O VI and O VII for UV and X-ray modeling

Aimed at ionization balance and spectral analysis of UV and X-ray sources, we present self-consistent sets of photoionization cross sections, recombination cross sections, and rate coefficients for Li-like O VI and He-like O VII. Relativistic fine structure is considered through the Breit-Pauli R-matrix (BPRM) method in the close coupling approximation, implementing the unified treatment for total electron-ion recombination subsuming both radiative and di-electronic recombination processes. Self-consistency is ensured by using an identical wavefunction expansion for the inverse processes of photoionization and photo-recombination. Radiation damping of resonances, important for H-like and He-like core ions, is included. Compared to previous LS coupling results without radiative decay of low-n (<= 10) resonances, the presents results show significant reduction in O VI recombination rates at high temperatures. In addition to the total rates, level-specific photoionization cross sections and recombination rates are presented for all fine structure levels n (lSLJ) up to n <= 10, to enable accurate computation of recombination-cascade matrices and spectral formation of prominent UV and X-ray lines such as the 1032,1038 A doublet of O VI, and the `triplet' forbidden, intercombination, and resonance X-ray lines of O VII at 22.1, 21.8, and 21.6 \ang respectively. Altogether, atomic parameters for 98 levels of O VI and 116 fine structure levels of O VII are theoretically computed. These data should provide a reasonably complete set of photoionization and recombination rates in collisional or radiative equilibrium.Comment: 33 pages, 8 figures, submitted to ApJ