In this chapter we study the problem of valuing a (possibly defaultable) derivative asset
contingent on the underlying economic state modeled as a Markov process. To gain …

Mathematical finance is an old science but has become a major topic for numerical analysts
since Merton [97], Black—Scholes [16] modeled financial derivatives. An excellent book for …

We consider discretizations of the heat equation by A-stable θ-schemes in time and
conforming finite elements in space. For these discretizations we derive residual a posteriori …

We derive a posteriori error estimates for fully discrete approximations to solutions of linear
parabolic equations. The space discretization uses finite element spaces that are allowed to …

This article is a review of basic concepts and tools devoted to a posteriori error estimation for
problems solved with the finite element method. For the sake of simplicity and clarity, we …

We derive a posteriori error estimates for the discretization of the heat equation in a unified
and fully discrete setting comprising the discontinuous Galerkin, various finite volume, and …

We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin
method dG (q) and the corresponding implicit Runge-Kutta-Radau method IRK-R (q) of …

We consider the a posteriori error analysis of approximations of parabolic problems based
on arbitrarily high-order conforming Galerkin spatial discretizations and arbitrarily high-order …

We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit
Euler in time) to coupled PDE systems in which one dependent variable, say u, is governed …

We derive energy-norm a posteriori error bounds for an Euler time-stepping method
combined with various spatial discontinuous Galerkin schemes for linear parabolic …