A new computational method was developed for modeling the effects of the geometric complexity, nonuniform muscle fiber orientation, and material inhomogeneity of the ventricular wall on cardiac impulse propagation. The method was used to solve a modification to the FitzHugh-Nagumo system of equations. The geometry, local muscle fiber orientation, and material parameters of the domain were defined using linear Lagrange or cubic Hermite finite element interpolation. Spatial variations of time-dependent excitation and recovery variables were approximated using cubic Hermite finite element interpolation, and the governing finite element equations were assembled using the collocation method. To overcome the deficiencies of conventional collocation methods on irregular domains, Galerkin equations for the no-flux boundary conditions were used instead of collocation equations for the boundary degrees-of-freedom. The resulting system was evolved using an adaptive Runge-Kutta method. Converged two-dimensional simulations of normal propagation showed that this method requires less CPU time than a traditional finite difference discretization. The model also reproduced several other physiologic phenomena known to be important in arrhythmogenesis including: Wenckebach periodicity, slowed propagation and unidirectional block due to wavefront curvature, reentry around a fixed obstacle, and spiral wave reentry. In a new result, the authors observed wavespeed variations and block due to nonuniform muscle fiber orientation. The findings suggest that the finite element method is suitable for studying normal and pathological cardiac activation and has significant advantages over existing techniques. >

A collocation-Galerkin finite element model of cardiac action potential propagation

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines. The analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications. Equations to which the analysis is applied include Fredholm integral equations of the second kind, certain first kind Fredholm equations, singular integral equations involving Cauchy kernels, a variety of integro-differential equations, and two-point boundary value problems for ordinary differential equations. The error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard Galerkin methods. We compare the collocation method with a standard Galerkin method using splines of the same degree, showing that the Galerkin method is quasioptimal in a Sobolev space of lower index and furnishes optimal order approximation for a range of Sobolev indices containing and extending below that for the collocation method, and so the standard Galerkin method achieves higher rates of convergence.

On the asymptotic convergence of collocation methods

A brief survey with a bibliography of superconvergence phenomena in finding a numerical solution of differential and integral equations is presented. A particular emphasis is laid on superconvergent schemes for elliptic problems in the plane employing the finite element method.

/pdf/on-superconvergence-techniques-5fxpdvilnw.pdf

On superconvergence techniques

Quadrature formulas of degrees 4 to 8 for numerical integration over the tetrahedron are constructed. The formulas are fully symmetric with respect to the tetrahedron, and in some cases are the minimum point rules with this symmetry.

Moderate-degree tetrahedral quadrature formulas

Tutorial introduction Algorithms of CM, TM, and CTM Coupling techniques Boundary element methods Other kinds of boundary methods Comparisons Part I: Collocation Trefftz method 1 - Basic algorithms and theory Notations and preliminaries Approximation problems Error estimates Debye-Huckel equation Stability analysis 2 - Motz's problem and its variants Introduction Basic algorithms of CTM Error bounds and integration approximation Variants of Motz's problem Concluding remarks 3 - Coupling techniques Introduction Description of generalized TMs Penalty TMs Simplified hybrid TMs Penalty plus hybrid TMs Lagrange multiplier TM Effective condition number Numerical experiments 4 - Biharmonic equations with singularities Introduction The Green formulas of A2u The collocation Trefftz methods Error bounds Part II: Collocation methods 5 - Collocation methods Introduction Description of collocation methods Error analysis Robin boundary conditions Inverse inequalities Final remarks 6 - Combinations of collocation and finite element methods Introduction Combinations of FEMs Linear algebraic equations of combination of FEM and CM Uniformly Vh0-elliptic inequality Uniformly Vh0-elliptic inequality involving integration approximation Final remarks 7 - Radial basis function collocation methods Introduction Radial basis functions Description of radial basis function collocation methods Inverse estimates for radial basis functions Numerical experiments Comparisons and conclusions Part III: Advanced topics 8 - Combinations with high-order FEMs Introduction Combinations of TM and Lagrange FEMs Global superconvergence Adini's elements 9 - Eigenvalue problems Introduction New numerical algorithms for eigenvalue problems Error bounds of eigenvalues Error bounds of eigenfunctions Computational models and numerical experiments Eigenvalues for the singularity problem Summaries and discussions 10 - The Helmholtz equation Introduction The Trefftz method Error analysis Summaries and discussions 11 - Explicit harmonic solutions of Laplace's equation Introduction Harmonic functions Harmonic solutions involving Neumann conditions Extensions and analysis on singularity New models of singularities for Laplace's equation Concluding remarks Appendix - Historic review of boundary methods Potential theory Existence and uniqueness Reduction in dimensions and Green's formula Integral equations Extended Green's formula Pre-electronic computer era Electronic computer era Boundary integral equation and boundary element methods

/pdf/trefftz-and-collocation-methods-1x7y3q2kmy.pdf

Trefftz and Collocation Methods

A collocation–Galerkin method for the two point boundary value problem based on continuous piecewise polynomial spaces is defined where the collocation points are the roots of a Jacobi polynomial.An existence and uniqueness theorem is derived by means of a related variational procedure that is defined using a semi-discrete innerproduct.Optimal rates of convergence are established and $O(h^{2r} )$ order of superconvergence at the nodes is obtained. Additional approximations to the solution and its derivatives that obtain $O(h^{2r} )$ order of superconvergence at any point are described.

A Collocation–Galerkin Method for the Two Point Boundary Value Problem Using Continuous Piecewise Polynomial Spaces

Methods of polynomial degree d to approximate integrals over the surface of the s-dimensional sphere are discussed. Such methods are constructed by choosing the abscissas and weights so that the formulas integrate a certain subset of the polynomials of maximum degree d, exactly. A simple set of monomials is described, the exact integration of which will ensure that the method has the required polynomial degree. Results are obtained which enable the derivation of consistency conditions on the rule structures. An example of a new s-dimensional degree-11 rule is given.

Fully Symmetric Integration Formulas for the Surface of the Sere in S Dimensions

A computational study of five finite element methods for the solution of a single second order linear ordinary differential equation subject to general linear, separated boundary conditions is described In each method, the approximate solution is a piecewise polynomial expressed in terms of a B-spline basis, and is determined by solving a system of linear algebraic equations with an almost block diagonal structure The aim of the investigation is twofold: to determine if the theoretical orders of convergence of the methods are realized in practice, and to compare the methods on the basis of cost for a given accuracy In this study three parametrized families of test problems, containing problems of varying degrees of difficulty, are used The conclusions drawn are rather straightforward Collocation is the cheapest method for a given accuracy, and the easiest to implement Also, for solving the linear algebraic equations, the use of a special purpose solver which takes advantage of the structure of the equations is advisable 1 Introduction Finite element methods for two-point boundary value problems for a single ordinary differential equation may take many forms The simplest type of method is probably collocation (3) where an approximate solution is sought in some finite space subject to the constraint that it satisfy the differential equation at certain specified points This type of method has been applied very successfully in the case of mixed order systems of boundary value problems; see, for example, (1) Other methods can be described based on the standard L2-Galerkin approximation (9), or on a combination of this and the collocation approach (7), (13), (25) In addition, different weak formulations of the boundary value problem lead to two other techniques, the Hx- and //"'-Galerkin methods (12), (14), (15), (17) How one would apply these methods to a system of differential equations, with the exception of the collocation case, is not immediately clear In addition, it is not apparent what the relative advantages are of these various methods, even when applied to a single equation In this paper we restrict our attention to the case of a single linear differential equation Our aim is to investigate numerically five finite element techniques for a second order linear two-point boundary value problem, with separated boundary conditions of a general nature In particular, we consider (i) the organization of the methods; (ii) the treatment of the boundary conditions;

A computational study of finite element methods for second order linear two-point boundary value problems

A collocation-$H^{ - 1} $-Galerkin procedure is introduced and analyzed for a one space dimensional parabolic partial differential equation. Special attention is devoted to the behavior of the error due to the time dependent nature of the coefficients. Global error estimates are derived.

Collocation-H({) - 1} -Galerkin Method for Parabolic Problems with Time Dependent Coefficients

Abstrac. A collocation-Galerkin scheme is proposed for an initial-boundary value problem for a first order hyperbolic equation in one space dimension. The Galerkin equations satisfied by the approximating solution are obtained from a weak-weak formulation of the initial-boundary value problem. The collocation points are taken to be affine images of the roots of the Jacobian polynomials of degree r 1 on [0, 11 with respect to the weight function x(l x). Optimal L'(L2)-norm estimates of the error are derived.

/pdf/a-collocation-galerkin-method-for-a-first-order-hyperbolic-589f9j66tn.pdf

Julio César Díaz

Papers

A Collocation–Galerkin Method for the Two Point Boundary Value Problem Using Continuous Piecewise Polynomial Spaces

Fully Symmetric Integration Formulas for the Surface of the Sere in S Dimensions

A computational study of finite element methods for second order linear two-point boundary value problems

Collocation-H({) - 1} -Galerkin Method for Parabolic Problems with Time Dependent Coefficients

A collocation-Galerkin method for a first order hyperbolic equation with space and time dependent coefficient