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Integrodifference equations are successful and popular models in theoretical ecology to describe spatial dispersal and temporal growth of populations with nonoverlapping generations. In relevant situations, such infinite-dimensional discrete dynamical systems have a globally attractive periodic solution. We show that this property persists under sufficiently accurate spatial (semi-) discretizations of collocation and degenerate kernel-type using linear splines. Moreover, convergence preserving the order of the method is established. This justifies theoretically that simulations capture the behavior of the original problem. Several numerical illustrations confirm our results. [ABSTRACT FROM AUTHOR]

In this paper we present an error analysis of an Eulerian finite element method for solving parabolic partial differential equations (PDEs) posed on evolving hypersurfaces in Rd, d = 2, 3. The method employs discontinuous piecewise linear in time-continuous piecewise linear in space finite elements and is based on a space-time weak formulation of a surface PDE problem. Trial and test surface finite element spaces consist of traces of standard volumetric elements on a space-time manifold resulting from the evolution of a surface. We prove first order convergence in space and time of the method in an energy norm and second order convergence in a weaker norm. Furthermore, we derive regularity results for solutions of parabolic PDEs on an evolving surface, which we need in a duality argument used in the proof of the second order convergence estimate. [ABSTRACT FROM AUTHOR]

In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the n-dimensional hypersurface, Γ ⊂ ℝn+1, is embedded in a polyhedral domain in ℝn+1 consisting of a union, Th, of (n + 1)-simplices. The unifying feature of the methodological approach is that the finite element approximating space is based on continuous piecewise linear finite element functions on the bulk triangulation Th which is independent of Γ. Our first method is a sharp interface method (SIF) which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, Γh, of Γ. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes SIF from the method of [M. A. Olshanskii, A. Reusken, and J. Grande, SIAM J. Numer. Anal., 47 (2009), pp. 3339-3358]. The second method is a narrow band method (NBM) in which the region of integration is a narrow band of width O(h). NBM is similar to the method of [K. Deckelnick et al., IMA J. Numer. Anal., 30 (2010), pp. 351-376] but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface L² and H¹ norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces, and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection-diffusion conservation law. Numerical results are given which illustrate the rates of convergence. [ABSTRACT FROM AUTHOR]

Polynomial interpolants defined using Chebyshev extreme points as nodes converge uniformly at a geometric rate when sampling a function that is analytic on an interval. However, the convergence rate can be arbitrarily close to unity if the function has a singularity close to the interval when extended to the complex plane. In such cases, splitting the interval and doing piecewise interpolation may be more efficient in the total number of nodes than the global interpolant. Because the convergence rate is determined by Bernstein ellipses obtained through a Joukowski conformal map, relative efficiency of splitting at any point in the interval can be calculated and then optimized over the interval. The optimal splitting may be applied recursively. The Chebfun software project uses a simple rule of thumb without prior singularity information to create a binary search that can be shown to do an excellent job of finding the optimal splitting in most cases. However, the process can use a large number of intermediate function evaluations that are not needed in the final approximant. A Chebyshev-Padé technique generates approximate singularity locations that are good enough to get close to the optimal splitting more directly in test cases. The technique is applied to a singularly perturbed boundary-value problem. [ABSTRACT FROM AUTHOR]

We derive Sobolev-type inner products with respect to which hat functions on arbitrary triangulations of domains in ℝd are orthogonal. Compared with linear interpolation, the resulting approximation schemes yield superior accuracy at little extra cost. [ABSTRACT FROM AUTHOR]