Abstract
The aim of this paper is to derive a posteriori error estimates for semilinear parabolic interface problems. More specifically, optimal order a posteriori error analysis in the - norm for semidiscrete semilinear parabolic interface problems is derived by using elliptic reconstruction technique introduced by Makridakis and Nochetto in (2003). A key idea for this technique is the use of error estimators derived for elliptic interface problems to obtain parabolic estimators that are of optimal order in space and time.
Highlights
Whilst the topic of a posteriori error estimation for linear and nonlinear parabolic problems is relatively well understood for both conforming and nonconforming methods, see, e.g., [1-15], there are comparatively few results for semi linear parabolic interface problems [16-21]
Gupta and Sinha [20] used elliptic reconstruction techniques to derive a posterior error estimates for semi linear parabolic interface problems with a locallyLipschtiz continuous nonlinearity in the forcing term. They used a Backward-Euler-Galerkin scheme to discretise in time with a conforming finite element method in space
This paper aims to derive an optimal order a posteriori error estimates in the L∞(L2) + L2(H1)
Summary
Whilst the topic of a posteriori error estimation for linear and nonlinear parabolic problems is relatively well understood for both conforming and nonconforming methods, see, e.g., [1-15], there are comparatively few results for semi linear parabolic interface problems [16-21]. Cangiani et al [17] used a nonstandard elliptic projection of Douglas and Dupont [16] to derive optimal order a priori error estimates for these problems in L¥ ( L2 )-norm and extended this work to the fully discrete setting in [18]. Metcalfe [19] derived optimal order a posteriori error estimates in the L∞(L2) + L2(H1) norm for fully discrete parabolic interface problems. Gupta and Sinha [20] used elliptic reconstruction techniques to derive a posterior error estimates for semi linear parabolic interface problems with a locallyLipschtiz continuous nonlinearity in the forcing term. They used a Backward-Euler-Galerkin scheme to discretise in time with a conforming finite element method in space. Sabawi [21] has derived an a posteriori error estimate for a class of nonlinear parabolic interface problems involving possibly curved interfaces, with flux balancing interface conditions, e.g., modelling mass transfer of solutes through semi-permeable membranes, in both the
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