A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne-Weinberger inequality

We consider evolutionary reaction-diffusion problem with mixed Dirichlet--Robin boundary conditions. For this class of problems, we derive two-sided estimates of the distance between any function in the admissible energy space and exact solution of the problem. The estimates (majorants and minorants) are explicitly computable and do not contain unknown functions or constants. Moreover, it is proved that the estimates are equivalent to the energy norm of the deviation from the exact solution.

1. Problem statement. Let Ω ∈ R d be a bounded connected domain with Lipchitz continuous boundary ∂Ω, which consists of two measurable non-intersecting parts Γ D and Γ R associated with the Dirichlet and Robin boundary conditions, respectively. By Q T we denote the space-time cylinder Q T := Ω × (0, T ), T > 0, and S T := ∂Ω × [0, T ] = Γ D ∪ Γ R × [0, T ]. The parts of S T related to Γ D and Γ R are denoted by S D and S R , respectively.
We consider the classical reaction-diffusion initial boundary value problem: find u(x, t) and p(x, t) such that p · n + σu = g, where n denotes the vector of unit outward normal to ∂Ω, and f (x, t) ∈ L 2 (Q T ), ϕ(x) ∈ L 2 (Ω), g(x, t) ∈ L 2 0, T ; L 2 (S R ) .
The function λ entering the reaction part of (1) is a non-negative bounded function, which values may vary from very small (or zero) to large values in different parts of the domain. The function σ(s, t) is a bounded function defined on Γ R . We assume that for any (x, t) ∈ Q T the matrix A is symmetric and satisfies the condition By · Ω and · QT , we denote the standard norms in L 2 (Ω) and L 2 (Q T ), respectively. L 2,1 (Q T ) is the space of functions g(x, t) with the finite norm The generalized solution of (1)-(4) is defined as a function u(x, t) ∈ V 1,0 2 (Q T ), satisfying the integral identity Classical solvability results (see, e.g., [4,5,3]) guarantee that u exists and is unique in V 1,0 2 (Q T ). Assume that v ∈H 1 (Q T ) is an approximation of u. Our goal is to deduce explicitly computable and realistic estimates of the distance between u and v. In other words, we wish to quantify neighborhoods of the exact solution in terms of local topology equivalent to the natural energy norm. More precisely, we introduce the measure where ν, θ, ζ and χ are certain positive weights (balancing different components of the error). They can be selected in different ways so that (8) presents a collection of different error measures. Here, henceforth, we also use the norms In Theorem 2.1, we derive a fully computable and guaranteed upper bound of e = u − v (for this purpose we use the method originally introduced in [12]). In [15], this method was applied to problems with convection, and in [9] guaranteed error majorants were derived for the Stokes problem. In Section 2, we combine this approach with the technique suggested in [14] for the stationary reaction-diffusion 3 problem, which yields efficient bounds of the distance to the exact solution (error majorants) for problems with strongly changing reaction function. The majorant presented in Theorem 2.1 contains the constant C FΩ in the Friedrichs type inequality (18). If S T = S D , then this constant (or a guaranteed upper bound of it) is easy to find. However, in the case of mixed boundary conditions and complicated domains, finding C FΩ may cause a serious problem. Therefore, in Theorems 2.2 and 2.3, we derive another upper bounds, which are based on decomposition of Ω into a collection of non-overlapping convex sub-domains. By means of a technique close to that has been used in [13] for elliptic problem, we deduce majorants, which involve only constants in the Poincare type inequalities. For convex domains these constants are easy to estimate due to the well known result of Payne and Weinberger [11] (with correction of Bebendorf [2]). Therefore, we obtain a fully computable error majorant (13), which involves only known data and constants. In Subsection 2.2, we prove that it is equivalent to the distance to the exact solution measured in terms of the combined (primal-dual) norm.
An advanced form of the majorant (which is sharper than those in Theorems 2.1, 2.2, and 2.3 but has a more complicated structure) is derived in Section 3. In Subsection 3.2, it is shown that the advanced majorant is equivalent to the distance to the exact solution measured in terms of the primal energy norm. A guaranteed and fully computable lower bound of the error is derived in Theorem 4.1. The minorant (87) also contains only known data and can be computed directly. Finally, we note that the practical efficiency of estimates similar to those derived in this paper has been recently tested and confirmed in [7].

2.
Majorants of the deviation from u. In this section, we deduce the first (and the simplest) form of the functional, which provides a guaranteed and fully computable upper bound of the deviation (error) e = u − v for any function v ∈H 1 (Q T ) and the solution u. From (7), it follows that Since e ∈H 1 (Q T ), we can set η = e, use the relation and obtain This relation is a form of the 'energy-balance' identity in terms of deviations. It plays an important role in subsequent analysis. Next, we introduce an additional Theorem 2.1. (i) For any v ∈H 1 (Q T ) and y ∈ Y * div (Q T ) the following inequality holds: C FΩ is the constant in the Friedrichs' inequality C tr is the constant in the trace inequality related to the Robin part of the boundary (ii) For any δ ∈ (0, 2], γ ≥ 1, and µ ∈ [0, 1], the lower bound of the variation problem generated by the majorant is zero, and it is attained if and only if v = u and y = A∇u. Proof. (i) We transform the right-hand side of (11) by means of the relation QT div y η dxdt + QT y · ∇η dxdt = SR y · n dsdt, 5 which yields where By means of the Hölder inequality, we find that and where ν 1 is the constant in (6). Let µ(x, t) be a real-valued function taking values in [0, 1]. Next, we estimate the term I f as follows: In [14], this decomposition was used in order to overcome difficulties arising in the stationary problem if λ is small (or zero) in some parts of the domain and large in another (more detailed study of this form of the majorant can be found in [10] and [6]). By combining (24)-(26), we obtain The second term in the right-hand side of (27) is estimated by the Young-Fenchel inequality where γ is an arbitrary positive constant parameter. Analogously, Here, α 1 (t), α 2 (t), and α 3 (t) are functions satisfying the relation (20). Then, the estimate (13) follows from (28) Thus, we see that M 2 I (u, A∇u; δ, γ, µ) = 0 and, therefore, the exact lower bound of M 2 I (v, y; δ, γ, µ) is attained on the pair presenting the exact solution of (1)-(4). Assume that M 2 I (v, y; δ, γ, µ) = 0, which means that for a.a. (x, t) ∈ Q T the following relations hold: In view of (32), this relation is equivalent to (7), whence it follows that v = u and y = A∇u.
Remark 1. We see that M 2 I (v, y; δ, γ, µ) depends on a collection of parameters, which can be selected within certain admissible sets. Varying δ and γ allows us to obtain estimates for different error measures. By selecting the functions α i and µ, we find the best possible value of the majorant. This fact is beneficial for practical applications because we can select values of the parameters in an optimal way for a concrete problem. In particular, µ can be set to 0 and 1. For these two cases, we use the abridged notation M 2 I, µ=0 and M 2 I, µ=1 : The majorant M 2 I, µ=0 is well adapted to problems, in which λ is small or zero (so that the impact of the reaction term is insignificant). In such type problems, we should avoid the term , which makes the whole estimate sensitive to the residual R f (v, y) and may lead to a considerable overestimation of the error. The estimate M 2 I, µ=1 is useful if λ is not small and may attain large values in some parts of Ω. If λ reaches both small (or zero) and large values, then the combined estimate (13) is preferable.

2.1.
Estimates based upon domain decomposition. The majorant defined by (13) contains the Friedrichs constant C FΩ and the trace constant C tr . If Ω has a complicated geometry, then finding these constants (or guaranteed bounds of them) may not be an easy task. Below we suggest the method, which allows to overcome this difficulty. It is based on domain decomposition and leads to the estimates with a different set of constants (a consequent discussion of this method for elliptic problems can be found in [13]).
Assume that Ω is decomposed into a set of sub-domains We use the Poincare inequalities where w = w− |w| Ωi , and |w| Ωi denotes the mean value of w on Ω i . If all Ω i are convex, then C PΩ i can be estimated from the above by the quantity diam Ω i /π (see [11]). We use this fact in order to represent the majorant in a somewhat different form. In further analysis, we assume (for the sake of simplicity only) that S T = S D and ϕ(x) = v(x, 0).

Theorem 2.2.
For any v ∈H 1 (Q T ) and y ∈ Y * div (Q T ) the following inequality holds: (15) and (16), respectively, and Here, λ, and ζ = 1, and α 1 (t), α 2 (t) are positive scalar-valued functions satisfying the relation Proof. Consider the integral identity (22). The term I f can be represented as I µ f is estimated as By means of the Hölder inequality, for I 1−µ f we have Each of the terms on the right-hand side of (43) can be estimated as follows: At last, using the Young-Fenchel inequality, we obtain the following estimates and, analogously, By combining (46)-(49), we obtain (40).
Consider a special case, which arises if we impose additional conditions, namely, where µ is inherited from (13). Since the functions y and µ are in our disposal, these integral type conditions do not lead to essential technical difficulties provided that N is not too large. Now, (40) can be represented in a simpler form.

S. MATCULEVICH AND P. NEITTAANMÄKI AND S. REPIN
By means of the Young-Fenchel inequality, we deduce The term I d is estimated analogously to the method used in proof of Theorem 2.1: Therefore, (54)-(56) yield the estimate (51).

2.2.
Two sided estimates for combined norms. In modern numerical methods (e.g., in various mixed finite element schemes) the approximations are generated for both primal and dual components of the solution. We note that this concept is perfectly motivated by physical arguments because primal and dual components often reflect physically meaningful parts of the solution. By following this idea, we now consider the solution of (1)-(4) as a pair (u, p) ∈ V 1,0 2 (Q T ) × Y * div (Q T ). In order to measure the deviation of the approximation (v, y) ∈H 1 (Q T ) × Y * div (Q T ) from (u, p), we use the combined primal-dual norm (1 + β)||| y − A∇v ||| 2 Since p = A∇u, we reform the right-hand side of (58) as follows: By using (1), we find that whereν =θ = (1 + β),ζ = 1 + 1 β C 2 FΩ ν1 , andχ = 1. Next, by combining the first two terms, applying (58), and, finally, adding and subtracting A∇v in the third term, we obtain Hence, we obtain the double inequality which shows that the majorant is equivalent to the combined primal-dual error norm. In other words, M 2 I (which contains only known functions and parameters) adequately reflects the distance from (v, y) ∈H 1 (Q T ) × Y * div (Q T ) to the exact solution (u, p). In particular, this means that if (u h , p h ) is the sequence of approximations computed on a certain set of meshes F h , which converges to (u, p) with the rate h α , then the values of the majorant tend to zero with the same rate.
(ii) This item is proven by the same arguments as in Theorem 2.1.

3.1.
An advanced majorant based upon domain decomposition. Now, we deduce an advanced versions of the estimates (40) and (51). Let (38) hold. First, we consider the case where λ is not small (or zero). Assume (for the sake of simplicity only) that S T = S D . Then, we have the following result.
Theorems 3.2 and 3.3 can be proven by combining arguments used in Theorems 2.2 and 2.3. Since proofs do not contain principally new items, we omit these details.

Equivalence of [e] 2
(ν, θ, ζ) and M 2 II . We aim to show that the advanced form of the majorant does not lead to an uncontrollable overestimation of the actual value of the norm (8). For this purpose, we estimate M 2 II from above and show that this upper bound is equivalent to the error norm. Henceforth, we assume that S T = S D , β = const and µ = 0. As before, these assumption are introduced for the sake of simplicity only. Similar estimates for the problems with mixed boundary conditions and variable coefficients can be deduced by arguments close to those presented below.
Proof. It is not difficult to see that we find that from one hand From another hand, (by using (7)) we see that for any η the functional generates the lower bound of the error norm defined in the right-hand side of the inequality (91).