On the Approximation of Nonlinear Evolution Equations in Particular C*-Algebras of Operators

Abstract

In this article we deal with the stability and convergence of numerical solutions of nonlinear evolution equations of the form $A(u(t))+f(u(t))=u'(t)$, the numerical analysis of solutions to this problems will be performed using some methods from particular algebras of operators which are sometimes represented by unital subalgebras of the unital C*-algebras of operators that are generated by some basic operators say $\mathbf{1},a,\mathcal{D}(\cdot)\in\mathcal{L}(H^m(G))$ that in some suitable sense are related to the operator $A(\cdot)\in\mathcal{L}(H^m(G))$ in the evolution equations, particular cases where the operator algebras do not verify the C*-identity with respect to the norm chosen are also studied, when applicable basic C*-algebra techniques are implemented to perform some estimates of numerical solutions to some types of problems, in all this work expressions like $H^m(G)$ will represent a prescribed discretizable Hilbert space with $G\subset\subset\mathbb{R}^n$.