Space-Time Isogeometric Method for a Nonlocal Parabolic Problem

TL;DR

This paper develops a space-time isogeometric method for solving a nonlocal parabolic problem, proving existence, uniqueness, and error estimates, and validating the approach with numerical experiments.

Contribution

It introduces a novel space-time isogeometric discretization for nonlocal parabolic equations, including theoretical analysis and practical linearization techniques.

Findings

Proven existence and uniqueness of solutions.

Established a priori error estimates.

Numerical experiments confirm theoretical results.

Abstract

In the present work, we focus on the space-time isogeometric discretization of a parabolic problem with a nonlocal diffusion coefficient. The existence and uniqueness of the solution for the continuous space-time variational formulation are proven. We prove the existence of the discrete solution and also establish the a priori error estimate for the space-time isogeometric scheme. The non-linear system is linearized through Picards method and a suitable preconditioner for the linearized system is provided. Finally, to confirm the theoretical findings, results of some numerical experiments are presented.