In [8], Gilbert Strang explored the chaotic behavior of Newton's method applied to x + 1 when the iteration starts from only real initial conditions. We embed Strang's problem in the one-parameter family of iteration methods known as relaxed Newton's method. By extending this analysis to include a discussion of invariant measures, probability densities, and Lyapunov exponents, we prove that in ...

Abstract. We develop a locally conservative Eulerian-Lagrangian finite volume scheme with the weighted essentially non-oscillatory property (EL-WENO) in one-space dimension. This method has the advantages of both WENO and Eulerian-Lagrangian schemes. It is formally high-order accurate in space (we present the fifth order version) and essentially non-oscillatory. Moreover, it is free of a CFL ti...

The Strang and Fix conditions relate the accuracy of a finite element method to its ability to reproduce polynomials. A similar condition is proved to exist for the approximation of the product on these finite elements. Piecewise polynomial approximations are studied further, including a constructive description of all related approximate product operators.

In this paper we consider an intrinsic approach for the direct computation of the fluxes for problems in potential theory. We develop a general method for the derivation of intrinsic conforming and non-conforming finite element spaces and appropriate lifting operators for the evaluation of the right-hand side from abstract theoretical principles related to the second Strang Lemma. The convergen...

We develop and demonstrate a computationally efficient numerical splitting technique for solving the reaction–diffusion equation. The scheme is based on the Strang splitting technique wherein the portions of the governing equations containing stiff chemical reaction terms are separated from those parts containing the less-stiff transport terms. As demonstrated, the scheme achieves second-order ...

We present a new method to construct interpolating reenable functions in higher dimensions. The approach is based on the solutions to speciic Lagrange interpolation problems by polynomials and applies to a large class of scaling matrices. The resulting scaling functions automatically satisfy certain Strang{Fix conditions. Several examples are discussed.

The Broyden class of quasi-Newton updates for inverse Hessian approximation are transformed to the formal BFGS update, which makes possible to generalize the well-known Nocedal method based on the Strang recurrences to the scaled limited-memory Broyden family, using the same number of stored vectors as for the limited-memory BFGS method. Two variants are given, the simpler of them does not requ...

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Semi-Lagrangian (SL) methods have been very popular in the Vlasov simulation community [28, 2, 3, 18, 31, 4, 23, 25]. In this paper, we propose a new Strang split SL discontinuous Galerkin (DG) method for solving the Vlasov equation. Specifically, we apply the Strang splitting for the Vlasov equation [5], as a way to decouple the nonlinear Vlasov system into a sequence of linear equations. To e...