Ismael Herrera

An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation

Abstract Many numerical methods use characteristic analysis to accommodate the advective component of transport. Such characteristic methods include Eulerian-Lagrangian methods (ELM), modified method of characteristics (MMOC), and operator splitting methods. A generalization of characteristic methods can be developed using an approach that we refer to as an Eulerian-Lagrangian localized adjoint method (ELLAM). This approach is a space-time extension of the optimal test function (OTF) method. The method provides a consistent formulation by defining test functions as specific solutions of the localized homogeneous adjoint equation. All relevant boundary terms arise naturally in the ELLAM formulation, and a systematic and complete treatment of boundary condition implementation results. This turns out to have significant implications for the calculation of boundary fluxes. An analysis of global mass conservation leads to the final ELLAM approximation, which is shown to possess the conservative property. Numerical calculations demonstrate the behaviour of the method with emphasis on treatment of boundary conditions. Discussion of the method includes ideas on extensions to higher spatial dimensions, reactive transport, and variable coefficient equations.

Trefftz Method: A General Theory

A precise definition of Trefftz method is proposed and, starting with it, a general theory is briefly explained. This leads to formulating numerical methods from a domain decomposition perspective. An important feature of this approach is the systematic use of “fully discontinuous functions” and the treatment of a general boundary value problem with prescribed jumps. Usually finite element methods are developed using splines, but a more general point of view is obtained when they are formulated in spaces in which the functions together with their derivatives may have jump discontinuities and in the general context of boundary value problems with prescribed jumps. Two broad classes of Trefftz methods are obtained: direct (Trefftz—Jirousek) and indirect (Trefftz—Herrera) methods. In turn, each one of them can be divided into overlapping and nonoverlapping. The generality of the resulting theory is remarkable, because it is applicable to any partial (or ordinary) differential equation or system of such equations, which is linear. The article is dedicated to Professor Jiroslav Jirousek, who has been a very important driving force in the modern development of Trefftz method.

Numerical modeling in science and engineering

Basic Equations of Macroscopic Systems. Introduction to Numerical Methods. Steady-state Systems. Dissipative Systems. Nondissipative Systems. High-order, Nonlinear, and Coupled Systems. Appendix: Summary of Vector and Tensor Analysis. Index.

Boundary methods, c-complete systems for stokes problems

A boundary method for solving Stokes problem is presented. This is based on the use of systems of solutions of the homogeneous equations, which are complete. A convenient criterion, for the completeness of such systems, is the notion of c-completeness. An apparently new representation of solutions of Stokes equations is derived and is used to develop a procedure for constructing a c-complete system. Examples of such systems are constructed.

Theory of connectivity: a systematic formulation of boundary element methods

Abstract A theory of connectivity recently developed by the author is applied to construct a systematic formulation of boundary element methods. The concept of complete connectivity condition is shown to supply an alternative to boundary integral equations. The general problem of connecting solutions defined in neighbouring regions R and E is shown to lead to complete connectivity conditions which permit the formulation of three kinds of variational principles; they involve, respectively, R ∪ E , R and the common boundary between R and E , only.

Scattering of SH waves by surface cavities of arbitrary shape using boundary methods

Abstract In this paper a boundary method is used to numerically solve the problem of scattering of SH waves by a bounded surface cavity or arbitrary shape in a half-space. This method reduces the dimension of the problem by one, but avoids the introduction of singular integral equations. A close connection is established between this method and least-squares collocation. Results are obtained using a multipole expansion in terms of Hankel functions about the origin. Comparison with some known exact solutions for SH wave motion yields very good agreement. It is observed that, in the case of a trench with steep walls, local amplification factors can sometimes significantly exceed 100%.

Boundary Methods. C-Complete Systems for the Biharmonic Equations

A boundary method for solving the biharmonic equation is presented. It is based on the use of systems of solutions of the homogeneous equations, which are complete. A convenient criterium for the completeness of such systems, is the notion of c-completeness. Using a convenient representation of solutions for the biharmonic equation a procedure for constructing c-complete systems for this equation is developed. Examples of such systems are constructed.

Trefftz-Herrera domain decomposition

Abstract The authors algebraic theory of boundary value problems has permitted systematizing Trefftz method and expanding its scope. The concept of TH-completeness has played a key role for such developments. This paper is devoted to revise the present state of these matters. Starting from the basic concepts of the algebraic theory, Green-Herrera formulas are presented and Localized Adjoint Method (LAM) derived. Then the classical Trefftz method is shown to be a particular case of LAM. This leads to a natural generalization of Trefftz method and a special class of domain decomposition methods: Trefftz-Herrera domain decomposition.

Indirect methods of collocation: Trefftz‐Herrera collocation

A nonstandard collocation method (TH-collocation) is presented, where collocation is used to construct specialized weighting functions instead of the solution itself, as it is usual, so that in this sense it is an indirect method. TH-collocation is shown to be as accurate as standard collocation, but computationally far more efficient. The present article is the first of a series devoted to explore thoroughly collocation methods. The following classification of collocation methods is introduced: direct-nonoverlapping; indirectnonoverlapping; direct-overlapping; and indirect-overlapping. Most of the effort reported in the literature has gone to direct-nonoverlapping methods. The procedure presented in this article falls into the indirectnonoverlapping category and it is based on Trefftz{Herrera formulation. c 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 709{738, 1999

Some Unifying Concepts in Applied Mathematics

I was trained, originally, as a pure mathematician although I have always worked in applications. Since I was a student I have recognized that the methodology of mathematical thinking is very powerful as a tool for development. It has, indeed, great practical value. Henry Pollak in his excellent talk today presented many interesting examples which seem to indicate that in the field of communications this is thoroughly recognized. However, there are still many areas of industry and of other human endeavors in which skepticism is prevalent.