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Francisco-Javier Sayas

Publications and source records attributed to Francisco-Javier Sayas.

44 records · Page 3Linked to original sources
arXiv2012

Convergence analysis of a high-order Nystrom integral-equation method for surface scattering problems

In this paper we present a convergence analysis for the Nystrom method proposed in [Jour. Comput. Phys. 169 pp. 2921-2934, 2001] for the solution of the combined boundary integral equation formulations of sound-soft acoustic scattering problems in three-dimensional space. This fast and efficient scheme combines FFT techniques and a polar change of variables that cancels out the kernel singularity. We establish the stability of the algorithms in the $L^2$ norm and we derive convergence estimates in both the $L^2$ and $L^\infty$ norms. In particular, our analysis establishes theoretically the previously observed super-algebraic convergence of the method in cases in which the right-hand side is smooth.

math.NA↗
arXiv2012

A decoupled preconditioning technique for a mixed Stokes-Darcy model

We propose an efficient iterative method to solve the mixed Stokes-Dracy model for coupling fluid and porous media flow. The weak formulation of this problem leads to a coupled, indefinite, ill-conditioned and symmetric linear system of equations. We apply a decoupled preconditioning technique requiring only good solvers for the local mixed-Darcy and Stokes subproblems. We prove that the method is asymptotically optimal and confirm, with numerical experiments, that the performance of the preconditioners does not deteriorate on arbitrarily fine meshes.

math.NA↗
arXiv2012

Strong coupling of finite element methods for the Stokes-Darcy problem

The aim of this paper is to propose a systematic way to obtain convergent finite element schemes for the Darcy-Stokes flow problem by combining well-known mixed finite elements that are separately convergent for Darcy and Stokes problems. In the approach in which the Darcy problem is set in its natural $\mathbf{H}(\text{div})$ formulation and the Stokes problem is expressed in velocity-pressure form, the transmission condition ensuring global mass conservation becomes essential. As opposed to the strategy that handles weakly this transmission condition through a Lagrange multiplier, we impose here this restriction exactly in the space of global velocity field. Our analysis of the Galerkin discretization of the resulting problem reveals that, if the mixed finite element space used in the Darcy domain admits an $\mathbf{H}(\text{div})$-stable discrete lifting of the normal trace, then it can be combined with any stable Stokes mixed finite element of the same order to deliver a stable global method with quasi-optimal convergence rate. Finally, we present a series of numerical tests confirming our theoretical convergence estimates.

math.NA↗
arXiv2012

A Note on the Ladyzenskaja-Babuska-Brezzi Condition

The analysis of finite-element-like Galerkin discretization techniques for the stationary Stokes problem relies on the so-called LBB condition. In this work we discuss equivalent formulations of the LBB condition.

math.NA↗
arXiv2011

Analysis of a non-symmetric coupling of Interior Penalty DG and BEM

We analyze a non-symmetric coupling of interior penalty discontinuous Galerkin and boundary element methods in two and three dimensions. Main results are discrete coercivity of the method, and thus unique solvability, and quasi-optimal convergence. The proof of coercivity is based on a localized variant of the variational technique from [F.-J. Sayas, The validity of Johnson-Nédeléc's BEM-FEM coupling on polygonal interfaces, {\em SIAM J. Numer. Anal.}, 47(5):3451--3463, 2009]. This localization gives rise to terms which are carefully analyzed in fractional order Sobolev spaces, and by using scaling arguments for rigid transformations. Numerical evidence of the proven convergence properties has been published previously.

math.NA↗
arXiv2009

Hilbert scales and Sobolev spaces defined by associated Legendre functions

In this paper we study the Hilbert scales defined by the associated Legendre functions for arbitrary integer values of the parameter. This problem is equivalent to study the left-definite spectral theory associated to the modified Legendre equation. We give several characterizations of the spaces as weighted Sobolev spaces and prove identities among the spaces corresponding to lower regularity index.

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