A Priori L2 Error Estimates for Galerkin Approximations to Semilinear Parabolic Integro-Differential Equations
Yanping Lin · 2023
In this paper we shall study Galerkin approximations to the solution of the nonlinear integro-differential equation of parabolic type: https://www.w3.org/1998/Math/MathML" display="block"> u t = ∇ ⋅ { a ( u ) ∇ u + ∫ 0 t b ( x , t , τ , u ( x , τ ) ) ∇ u ( x , τ ) d τ } + f ( u ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003420064/603a9f47-0abb-45de-9779-2fd4fbd766b6/content/eqn1443.tif "/> A so-called non-classical H 1 projection will be defined and used in the derivations of the optimal L 2 error estimates for the continuous, Crank-Nicolson and extrapolated Crank-Nicolson approximations.