Computable error bounds for collocation methods

Direct Error Bounds for Symmetric Rbf Collocation

The standard error bounds for interpolation by kernels or radial basis functions are generalized to symmetric PDE collocation problems. This involves generalized Power Functions, and these can be explicitly calculated. Final error bounds then are a product of Power Function values and the norm of the solution in the native space of the kernel. Since the latter can also be estimated well, the wh...

Computable backward error bounds for basic algorithms in linear algebra

Standard error estimates in numerical linear algebra are often of the form γk|R||S| where R,S are known matrices and γk := ku/(1−u) with u denoting the relative rounding error unit. Recently we showed that for a number of standard problems γk can be replaced by ku for any order of computation and without restriction on the dimension. Such problems include LUand Cholesky decomposition, triangula...

A Posteriori Error Estimates including Algebraic Error: Computable Upper Bounds and Stopping Criteria for Iterative Solvers

We consider the finite volume and the lowest-order mixed finite element discretizations of a second-order elliptic pure diffusion model problem. The first goal of this paper is to derive guaranteed and fully computable a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. We show that the algebraic error can be simply bounded using t...

A Posteriori Error Bounds for Meshless Methods

We show how to provide safe a–posteriori error bounds for numerical solutions of well-posed operator equations using kernel–based meshless trial spaces. The presentation is kept as simple as possible in order to address a larger community working on applications in Science and Engineering. 1 Operator Equations Most contributions within application–oriented conferences only present numerical res...

Computable Bounds for Polynomial Ergodicity