# Research Articles

The mathematical ideas leading to deltaBEM have have been
under development for a number of years and are still the
object of ongoing research. The paper

**
Domínguez, Víctor; Lu, Sijiang L.; Sayas,
Francisco-Javier. A Nyström flavored Calderón
calculus of order three for two dimensional waves,
time-harmonic and transient. Comput. Math. Appl.
67 (2014), no. 1, 217–236. **

contains the main elements of deltaBEM applied to the Helmholtz equation. Some theoretical arguments justifying the methods are given in this paper. An older version of the discrete calculus, of order two only, is given in

**Domínguez, Víctor; Lu, Sijiang; Sayas, Francisco-Javier.**

*A fully discrete Calderón calculus for two dimensional time harmonic waves.*Int. J. Numer. Anal. Model. 11 (2014), no. 2, 332–345.The extension to time harmonic linear elasticity is addressed in

**Dominguez, Victor; Sanchez-Vizuet, Tonatiuh; Sayas, Francisco-Javier.**

*A fully discrete Calderon Calculus for the two-dimensional elastic wave equation.*Comput. Math. Appl. 69 (2015) 620-635A comprehensive introduction to the algorithmic aspects of Convolution Quadrature (both the linear multistep and Runge-Kutta cases) is given in

**Hassell,Matthew; Sayas, Francisco-Javier.**

*Convolution Quadrature for Wave Simulations.*(To appear in SEMA-SIMAI Springer Series.)For more theoretical results, consult the references in the previous papers, and wait for a full forthcoming analysis of the entire collection of discrete operators. Preprints of the published papers linked above can be downloaded from arXiv.