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An -estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space.
In this work, we present an introduction to automatic differentiation,
its use in optimization software, and some new potential usages. We
focus on the potential of this technique in
optimization. We do not dive deeply in the intricacies of automatic
differentiation, but put forward its key ideas. We sketch a survey, as
of today, of automatic differentiation software, but warn the reader
that the situation with respect to software evolves rapidly. In the
last part of the paper, we present some...
The time-ordered exponential of a time-dependent matrix is defined as the function of that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in . The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by . Yet, the existence of such inverses, crucial to...
A. Cordero et. al (2010) considered a modified Newton-Jarratt's composition to solve nonlinear equations. In this study, using decomposition technique under weaker assumptions we extend the applicability of this method. Numerical examples where earlier results cannot apply to solve equations but our results can apply are also given in this study.
We present a local convergence analysis of a one parameter Jarratt-type method. We use this method to approximate a solution of an equation in a Banach space setting. The semilocal convergence of this method was recently carried out in earlier studies under stronger hypotheses. Numerical examples are given where earlier results such as in [Ezquerro J.A., Hernández M.A., New iterations of -order four with reduced computational cost, BIT Numer. Math. 49 (2009), 325–342] cannot be used to solve equations...
We analyse the error of interpolation of functions from the space in the nodes of a regular quadratic Lagrange finite element in 1D by interpolants from the local function space of this finite element. We show that the order of the error depends on the way in which the mutual positions of nodes change as the length of interval approaches zero.
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