Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data

Optimal error estimates of mixed FEMs for second order hyperbolic integro-differential equations with minimal smoothness on initial data

“…These results have recently been improved by Goswami et al. [14], who have established optimal convergence rate O(h 2 t −1 ) for u(t) in L 2 -norm and O(ht −1 ) for σ(t) L 2 -norm for all t ∈ (0, T ], when u 0 ∈ L 2 (Ω). Similar results have been obtained for the extended finite element method described below.…”

“…Essentially, integrating in time leads to a first order evolution process and hence, is instrumental in reducing the regularity requirements on the solution. Further, due to the presence of the integral term, it is observed that the concept of mixed Ritz-Volterra projections used earlier in [11], [12], [21] and [14] plays a crucial role in our analysis. For the completely discrete scheme, a major difficulty associated with a use of the nonstandard energy formulation by Baker is the presence of a fourth order time derivative of the displacement u in the bounds, see [9].…”

“…Proof. Consider the auxiliary elliptic problem: 14) and set p = ∇ζ and ψ = Ap so that ∇ · ψ = η u . Then, from the elliptic regularity result (2.2), it follows that…”

“…However, optimal rate O(h 2 t −1 ) for u is only established for a class of problems when A = aI and B = b(t, s)I, where a and b are independent of spatial variable x. These results have recently been improved by Goswami et al [14], who have established optimal convergence rate O(h 2 t −1 ) for u(t) in L 2 -norm and O(ht −1 ) for σ(t) L 2 -norm for all t ∈ (0, T ], when u 0 ∈ L 2 (Ω). Similar results have been obtained for the extended finite element method described below.…”

“…In the first part of this paper, an extended mixed method for (1.1) is proposed and analyzed. Such a method was analysed earlier in [7] and [1] for elliptic problem, [23] for degenerate nonlinear parabolic problem, [14] for parabolic integro-differential equations and references cited in. To motivate this new mixed method, we introduce two variables:…”