**(in construction. Updated in July 2017)**

## Topics

My research is focused on Numerical Analysis, mainly on the numerical solution of integral equations. Specifically:

- Analytic and Numerical analysis of periodic pseudodifferential equations
- Boundary integral methods
- Coupling of Boundary and Finite Element methods for solving elliptic problems.
- Scattering of acoustic waves. Design of robust BEM for solving these problems.
- Quadrature rules for highly oscillatory integrals

A Matlab code for the quadrature rules can be found in Clenshaw-Curtis rules for oscillatory integrals with logarithmic singularities

## Some people involved

Let me cite here briefly the people I’ve collaborated with: Francisco-Javier (Pancho) Sayas and his team, my former supervisor, now at University of Delaware, Ivan G. Graham and Valery Smyshlyaev, at University of Bath, Oscar P. Bruno, in CALTECH, Catalin Turc and Yassine Boubendir, at NJIT and Mahadevan Ganesh, from Colorado School of Mines.

## Papers

### Submitted

### To appear

- Boubendir, Y., Domínguez, V., Levadoux, D. and Turc, C.
*Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains*. Journal of Integral Equations and Applications. Preprint available in https://arxiv.org/abs/1510.06682

### 2016

- Barrios, T., Bustinza, R. and Domínguez, V.
*On the discontinuous Galerkin method for solving boundary Problems for the Helmholtz equation: A priori and a posteriori error analyses*. Journal of Computational and Applied Mathematics 300 (2016), pp. 312-340. Preprint available in http://arxiv.org/abs/1310.2847 - Boubendir, Y., Domínguez, V., Levadoux, D. and Turc, C.
*High-order Nyström discretizations for the solution of integral equation formulations of two-dimensional Helmholtz transmission problems*. IMA Journal on Numerical Analysis 36 (2016), pp. 463-492. Preprint available in https://arxiv.org/abs/1304.7218 - Domínguez, V. and Ganesh, M.
*Sobolev estimates for constructive uniform-grid FFT interpolatory approximations of spherical functions*. Advances in Computational Mathematics 42 -4 (2016), pp 843-887. Preprint available in https://arxiv.org/abs/1407.5152

### 2015

- Boubendir, Y., Domínguez, V., Levadoux, D. and Turc, C.
*Regularized combined field integral equations for acoustic transmission problems*. SIAM Journal on Applied Mathematics 75 -3 (2015), pp. 929 – 952. Preprint available in https://arxiv.org/abs/1312.6598 - Domínguez, V., Sánchez-Vizuet, T., and Sayas, F.J.
*A fully discrete Calderón calculus for the two-dimensional elastic wave equation*. Computers and Mathematics with Applications. 69 – 7 (2015), pp. 620 – 635. Preprint available in https://arxiv.org/abs/1210.7017

### 2014

- Domínguez, V.
*Filon-Clenshaw-Curtis rules for a class of highly-oscillatory integrals with logarithmic singularities*. Journal of Computational and Applied Mathematics , 261 (2014), pp 299–319. Preprint available in http://arxiv.org/abs/1305.1365 - Domínguez, V, Lu, S.L., and Sayas, F.J.
*A Nyström method for the two dimensional Helmholtz hypersingular equation*. Computers and Mathematics with Applications 67 (2014) pp. 217–236. Preprint available in https://arxiv.org/abs/1210.4582 - Domínguez, V, Lu, S.L., and Sayas, F.J. A
*Nystrom flavored Calderón Calculus of order three for two dimensional waves, time-harmonic and transient.*Computers and Mathematics with Applications 67 (2014), no. 1, 217–236. Preprint available in http://arxiv.org/abs/1304.7218 - Domínguez, V, Lu, S.L., and Sayas, F.J.
*A fully discrete Calderon Calculus for two dimensional time harmonic waves*. International Journal of Numerical Analysis & Modeling 11 (2014), 332-345 . Preprint available in http://arxiv.org/abs/1210.7017

### 2013

- Domínguez, V and Sayas, F.J.
*Some properties of layer potentials and boundary integral operators for the wave equation*. Journal of Integral Equations and Applications 25 (2013) no 2, pp. 253 – 294. Preprint available in http://arxiv.org/abs/1109.6352 - Domínguez, V and Ganesh, M.
*Interpolation and cubature approximations and analysis for a class of wideband integrals on the sphere*. Advances in Computational Mathematics 39 (2013) no 3-4, 547 -584. Preprint available in http://arxiv.org/abs/1204.5109 - Bruno, O.P., Domínguez, V, and Sayas, F.J.
*Convergence analysis of a high-order Nystrom integral-equation method for surface scattering problems*. Numerische Mathematik 124 (2013), no. 4, 603-645. Preprint available in http://arxiv.org/abs/1109.6352. - Domínguez, V, Graham, I.G., and Kim, T.
*Filon-Clenshaw-Curtis rules for highly-oscillatory integrals with algebraic singularities and stationary points*. SIAM Journal of Numerical Analysis 51 (2013) no. 3, 1542–1566. Preprint available in http://arxiv.org/abs/1207.2283

### 2011

- Domínguez, V., Graham, I., and Smyshlyaev, V.
*Stability and error estimates for Filon-Clenshaw-Curtis rules for highly-oscillatory integrals*. IMA Journal of Numerical Analysis 31 (2011), no 4, 1253-1280. Preprint available http://opus.bath.ac.uk/26655 - Domínguez, V., Heuer, N., and Sayas, F-J.
*Hilbert scales and Sobolev spaces defined by associated Legendre functions.*J. Comput. Appl. Math. 235 (2011), no. 12, 3481–3501. Preprint available in http://arxiv.org/abs/0909.4266.

### 2009

- Domínguez, V. and Sayas, F.-J. Algorithm 884: a simple Matlab implementation of the Argyris element. ACM Trans. Math. Software 35 (2009), no. 2, Art. 16, 11 pp. Preprint available here.

### 2008

- Boal, N.; Domínguez and V.; Sayas, F.-J.
*Asymptotic properties of some triangulations of the sphere.*J. Comput. Appl. Math. 211 (2008), no. 1, 11–22. Paper is here - Domínguez, V., Rapún, M.-L., and Sayas, F.-J.
*Dirac delta methods for Helmholtz transmission problems.*Adv. Comput. Math. 28 (2008), no. 2, 119–139.

### 2007

- Domínguez, V., Graham, I. G., and Smyshlyaev, V. P.
*A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering.*Numer. Math. 106 (2007), no. 3, 471–510. - Domínguez, V. and Sayas, F.-J.
*Overlapped BEM-FEM for some Helmholtz transmission problems.*Appl. Numer. Math. 57 (2007), no. 2, 131–146.

### 2006

- Domínguez, V. and Sayas, F.-J.
*A BEM-FEM overlapping algorithm for the Stokes equation.*Appl. Math. Comput. 182 (2006), no. 1, 691–710. - Domínguez, V. and Sayas, F.-J.
*Overlapped BEM-FEM for some Helmholtz transmission problems.*Appl. Numer. Math. 57 (2007), no. 2, 131–146.

### 2004

- Celorrio, R., Domínguez, V., and Sayas, F.-J.
*Overlapped BEM-FEM and some Schwarz iterations. Comput.*Methods Appl. Math. 4 (2004), no. 1, 3–22.

### 2003

- Domínguez, V. and Sayas, F.-J.
*Stability of discrete liftings. C. R. Math. Acad. Sci. Paris*337 (2003), no. 12, 805–808. - Domínguez, V.
*High-order collocation and quadrature methods for some logarithmic kernel integral equations on open arcs.*J. Comput. Appl. Math. 161 (2003), no. 1, 145–159.

### 2002

- Celorrio, R., Domínguez, V., and Sayas, F.-J.
*Periodic Dirac delta distributions in the boundary element method.*Adv. Comput. Math. 17 (2002), no. 3, 211–236. - Celorrio, R., Domínguez, V., and Sayas, F.-J.
*An interior-exterior Schwarz algorithm and its convergence.*C. R. Math. Acad. Sci. Paris 334 (2002), no. 10, 923–926.

### 2001

- Domínguez, V. and Sayas, F.-J.
*Local expansions of periodic spline interpolation with some applications.*Math. Nachr. 227 (2001), 43–62. - Domínguez, V. and Sayas, F.-J.
*Full asymptotics of spline Petrov-Galerkin methods for some periodic pseudodifferential equations.*Adv. Comput. Math. 14 (2001), no. 1, 75–101.

### Just one more thing

An implementation of the Clenshaw-Curtis rules for highly oscillatory integrals (see here, here and here) is gonna be back asap