)–Error Bounds in Discontinuous Galerkin Methods For Semidiscrete Semilinear Parabolic Interface Problems

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 L∞(L2) + L2(H )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.

More recently, 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 ∞ ( 2 ) + 2 ( 1 ) and L ¥ (L 2 )norms. Optimal order a posteriori error estimates ∞ ( 2 ) + 2 ( 1 )were derived for semi and fully discrete nonlinear parabolic interface problems.
The analysis revolved around a nonstandard elliptic reconstruction introduced by Douglas and Dupont (16).
The main contribution of this paper is to extend (21) to the case of semidiscrete semilinear parabolic interface problems in terms of ∞ ( 2 ) + 2 ( 1 )-norm. The main difficulty in constructing an optimal order a posteriori error estimator in ∞ ( 2 ) + 2 ( 1 )is to deal with the nonlinear reaction term. These challenges are addressed by employing a continuation argument and the elliptic reconstruction technique, introduced by Makridakis and Nochetto (12) and extended to dG methods in (7).
It is worth noting the main reason of this technique is to lead us utilise ready elliptic interface a posteriori estimates that derived from elliptic interface problem (22)(23) to bound the main part of the spatial error. There are some error estimators for semilinear parabolic problems available in the literature (24)(25)(26)(27)(28)(29)(30).
The rest of this paper is structured as follows. In Section 2, the model problem is introduced and discontinuous Galerkin method, with some necessary background results, are discussed. Section 3, ∞ ( 2 ) + 2 ( 1 )error bounds for semi discrete semilinear parabolic interface problems, are presented. Conclusions are given in Section 4.

Multiply (1) by a test function
∈ ℋ 0 1 and integration by parts on each sub-domain and applying the interface condition in (1), such that
where ( ) denotes the space of polynomials of total degree on an element . Suppose that 1 and 2 are two elements sharing the same face ⊂ Γ ∪ Γ , where ⊂ 1 ∩ 2 with 1 and 2 denoting the outward unit normal vectors on of 1 and 2 , respectively. Then, subdividing Γ the mesh skeleton into three disjoint subsets Γ = Ω ∪ Γ ∪ Γ , where Γ = Γ \( Ω ∪ Γ ) is the interior points. Let be a discontinuous function across Γ. Setting = | and defining its jump ⟦ ℎ ⟧ and average { ℎ } across by Similarly, for a vector valued function ℎ , piecewise smooth on with = | , such that Thus, setting { ℎ } = , ⟦ ℎ ⟧ = ⟦ ℎ ⟧ = ℎ • with denoting the outward unit normal on the boundary Ω. For the mesh size, using the function ℎ: Ω → ℝ, where ℎ| = ℎ , ∈ and ℎ = {ℎ} on each ( − 1) dimensional open face ⊂ Γ. Further, assuming that ℎ : = ∈Ω ℎ and ℎ : = ∈Ω ℎ. Without loss of generality, assuming that ℎ remains uniformly bounded throughout this work, to avoid having estimation constants depend on {1, ℎ }. To introduce the interior penalty discontinuous Galerkin method, Multiplying (1) by a test function in ∈ ℋ 0 1 + ℎ ( ) and, integrate over each subdomain, so that Then, splitting the integral into element contributions and integrate by parts: The next step is to decompose the face integrals: where K 1 and K 2 ∈ , for ∈ 1 ∩ 2 is a corresponding unit normal on (exterior to 2 ). Finally, it is ready to introduce the interior penalty discontinuous Galerkin method for (3), which reads: find ℎ ∈ ℎ such that ⟨ ℎ , ℎ ⟩ + ℎ ( ; ℎ , ℎ ) = ⟨ ( ℎ ), ℎ ⟩, where 0 and donate by the discontinuity penalization parameter and permeability coefficient, respectively. Here, 0 has to be chosen large enough to ensure the stability of the discontinuous Galerkin scheme.
Then, extending the norm ||| ||| to functions in ∈ ℋ 0 1 + ℎ ( ) by setting  To simplify the left-hand side and using the identity Therefore, Applying Cauchy-Schwarz inequality on the righthand side of above equation, this becomes The first term on the right-hand side of (11) can be handled using (2) as follows Hence, valid for all 0 £ r < 2for 2 d  and 0 £ r £ 4 3 for 3 d  . Substituting this into our earlier inequality (12), this leads to Now, putting ( ) = √1 + 4 | ℎ | ∞ 2 , this becomes Refuring (11)   To deal with final term on the right-hand side of (17), applying the inequality 1 2 ≤ ( 1 + 2 ) 1+ 1 and assume that the mesh-size ℎ is small enough that Since the left-hand side of (17) depends continuously on t, which implies that the set non-empty and closed. Therefore, setting * = max and assuming that * < , so that and using Gronwall's inequality, such that This leads to contradiction with hypothesis * < because of the continuity of the left-hand side of (19). Hence, * = . Setting * = , and ‖ ( * )‖ = ‖ ‖ ∞(0, ; (Ω)) due to the continuity with respect to , (19) implies The result therefore, follows from the last three inequalities.

Conclusion:
This paper aims to derive an optimal order a posteriori error estimates in the ∞ ( 2 ) + 2 ( 1 )norm for semdiscrete semilinear parabolic interface problems. An important factor in our analysis to derive this estimator is to use the elliptic reconstruction framework of Makridakis and Nochetto (12) although, crucially, a number of challenges have to be overcome due to non-linearity on the forcing term depending on Gronwall's Lemma and Sobolev embedding through continuation argument. The main use for these bounds is in designing an efficient adaptive scheme, and consequently leading to a reduction in the computational cost of the scheme. In the future, This work can be extended to the fully discrete case for semilinear parabolic interface problems in L ¥ (L 2 ) and L ¥ (H 1 ) norms.