A posteriori analysis of the spectral element discretization of a non linear heat equation

The a posteriori analysis technique presents a very e cient tool for themesh adaptivitymethods. Thesemethods have been widely applied in the context of the nite element discretization (see [2, 4, 5, 14, 17–20]). However, few works considered the discretization by the spectral method (see [1, 8, 10]). This paper dealswith the discretization of a non linear heat equation.Weuse Euler’s implicit schemewith respect to time and spectral element method with respect to space. The spectral element method consists in approaching the solution of a partial di erential equation by polynomial function of high degree on each sub-domain of a decomposition. This work is an extension of the results obtained by Bernardi et al. (see [8]) for a discretization based on the nite element method and Chor et al. (see [10]) for the case of a linear heat equation. Herein, we start by proving that the time semi discrete problem has a unique solution. We de ne two local families of residual error indicators (see [12]). A rst family related to the time discretization and depends only on the discrete solution and the time step. The value of this indicator allows us to choose the next time step. A second family of error indicators concerns the spectral discretization and explicitly depends on the discrete solution and the data of the non linear heat equation. We prove the optimality of those indicators in the sense that their Hilbertian sum is upper and lower bounded by the error estimation with constants independent of the discrete parameter in space and time. The paper is organized as follows: • In Section 2, we present the studied non linear heat equation. We prove the existence and uniqueness of the solution of the discrete time and full spectral problems. • Section 3 is related to the de nition of error indicators deduced from the residue of the non linear heat equation. We prove the equivalence between the error estimation and the Hilbertian sum of those error indicators.


Introduction
The a posteriori analysis technique presents a very e cient tool for the mesh adaptivity methods. These methods have been widely applied in the context of the nite element discretization (see [2,4,5,14,[17][18][19][20]). However, few works considered the discretization by the spectral method (see [1,8,10]). This paper deals with the discretization of a non linear heat equation. We use Euler's implicit scheme with respect to time and spectral element method with respect to space. The spectral element method consists in approaching the solution of a partial di erential equation by polynomial function of high degree on each sub-domain of a decomposition.
This work is an extension of the results obtained by Bernardi et al. (see [8]) for a discretization based on the nite element method and Chor et al. (see [10]) for the case of a linear heat equation. Herein, we start by proving that the time semi discrete problem has a unique solution. We de ne two local families of residual error indicators (see [12]). A rst family related to the time discretization and depends only on the discrete solution and the time step. The value of this indicator allows us to choose the next time step. A second family of error indicators concerns the spectral discretization and explicitly depends on the discrete solution and the data of the non linear heat equation. We prove the optimality of those indicators in the sense that their Hilbertian sum is upper and lower bounded by the error estimation with constants independent of the discrete parameter in space and time.
The paper is organized as follows: • In Section 2, we present the studied non linear heat equation. We prove the existence and uniqueness of the solution of the discrete time and full spectral problems.
• Section 3 is related to the de nition of error indicators deduced from the residue of the non linear heat equation. We prove the equivalence between the error estimation and the Hilbertian sum of those error indicators.

A time and space discrete problems
Let Ω be a bounded and simply connected domain of R d (d = , or ), ∂Ω is its connected Lipschitz continuous boundary and T is a positive constant. We consider the non linear heat equation: Find φ solution of in Ω. (2.1) We suppose that λ is a function verifying < m λ ≤ λ(x) ≤ M λ and |λ(x) − λ(y)| ≤ κ λ |x − y|, (2.2) f and φ are given functions and φ is an unknown function.
In order to study the variational formulation of problem (2.1), we de ne the following spaces • H s (Ω) s ≥ , is the Sobolev space provided with the norm || . || s,Ω and the semi-norm | .
We de ne the following norm: for any φ ∈ L ( , T, H (Ω)), . Time discretization To de ne the time semi discrete problem, we begin by introducing a partition of the interval [ , T] in sub- For the family (φ k ) ≤k≤K = φ(., t k ), we associate the function φτ, de ned on [ , T], a ne on each sub- By applying the Euler implicit method, we deduce the following time semi-discrete problem: in Ω, (2.6) where f k = f (., t k ). Problem (2.6) is equivalent to the following variational formulation: for f ∈ C ( , T, H − (Ω)) and φ ∈ L (Ω), nd (φ k ) ≤k≤K ∈ L (Ω) × (H (Ω)) K such that for any ≤ k ≤ K and ψ ∈ H (Ω): Proof 1. We begin by proving the existence of solution using Brouwer xed point theorem ( [13], chap 7). For ≤ k ≤ K, supposing φ k− is known, we de ne the application ϕ k , from H (Ω) into H (Ω), such that for φ k ∈ H (Ω) and ψ ∈ H (Ω), where (./.) is the scalar product in H (Ω).
Since λ is bounded, we conclude that ϕ k is continuous in H (Ω) and veri es for all φ k ∈ H (Ω) that Then, we have Then, (ϕ(φ k )/φ k ) is non negative on the sphere of H (Ω) with radius Since the sequence (φ k n ) is bounded in H (Ω), there exists a sub-sequence (φ k np ) which weakly converges to φ k in H (Ω). Consequently and according to the properties of the function λ (see (2.2)), we have: for each θm in Xm We conclude that lim The convergence of the remaining terms in (2.11) is easy to prove since they are linear. By the density of ∪ ∞ m= Xm in H (Ω), we deduce that φ k is solution of problem Showing now the stability condition (2.9): Let ψ = φ j in (2.7), then, Doing sum on j from to k, we conclude the stability condition (2.9).

De nition 2.1. We de ne the "local" norm on each ψ k in H (Ω) by
(2.14) Following the de nition of φτ, we deduce that Using inequality ab ≥ − a − b , we deduce To show the other inequality, we use the property ab ≤ a + b . We obtain For k = , we keep inequality (2.15). For k ≥ we use the following inequality Doing the sum on k, we conclude (2.14).

. Spectral element discretization
Since the polynomials inverse inequalities are not optimal in dimension d ≥ , herinafter, we consider only the one dimensional case for the a posteriori analysis of the spectral element method applied to the non linear heat equation. We start by describing the discrete problem deduced from the problem (2.7)-(2.8). Let Let N i an integer greater than , associated to the sub-domain Ω i , we de ne the discrete parameter where N an integer greater than and L N the Legendre polynomial de ned on Ω. We recall the following Gauss-Lobatto quadrature formula: where ρ N j , ≤ j ≤ N, represent the weights. Let Pn(Ω) the space of polynomial of degree ≤ n. We de ne in L (Ω) the discrete scalar product: For any continuous functions φ and ψ on Ω we recall the following property (see [7] for its proof) We consider i δ the Lagrange interpolation operator such that for φ continuous onΩ i ; Let the discrete space Then, the full discrete problem is: where λ δ is de ned, for each φ continuous onΩ, by The proof of the above theorem follows exactly the same idea as the proof of Theorem 2.1 by using Brouwer's xed point Theorem. We simply adjust the discrete norm to the continuous norm using inequality (2.18).
We refer to ( [16], chap 13) for the a priori analysis of the nite element discretization of this type of problems when the triangulations are independent of time.

Error indicators, lower and upper bounds
This section deals with the de nition of the two families of error indicators. The rst indicator is related to time discretization and the second one to spectral element discretization. We prove the equivalence of those indicators with the error estimate.

. The error indicators
The time error indicators are de ned by analogy to our previous work in the linear case (see [10,12]). For each k, ≤ k ≤ K, We also de ne the local indicators, which can be computed explicitly as a function of the discrete solution : For each k, ≤ k ≤ K and each sub-interval Ω i For technical reasons related to forthcoming demonstrations, we de ne the following discrete space:

. An upper bound for the error
Hereinafter, to upper-bound the error de ned with the norm introduced in (2.5) by the errors indicators and the data function, we apply the triangular inequality : (3.4) Proof 3. By replacing φ = φτ in (2.4), we obtain, for any ψ ∈ H (Λ) and t ∈ [t k− , t k ], Then, considering equation (2.6), The di erence between equation (3.5) and (2.4) results in: Since we have, for any u and v, and considering ψ = (φ − φτ), we obtain: Then, from the de nition of φτ, we have:

Let Cs the injection norm of the space H (Ω) in L p * (Ω). Then using inequality (3.3), we obtain:
By the triangular inequality and (3.9), we obtain: Doing the sum on m, we conclude that there exists a positive constant C, depending only on T, m λ , M λ , κ λ and γ, such that: (3.10) The property of norms equivalence (2.14) of lemma 2.1, yields that: where C is a positive constant independent of N.

Proposition 3.2. Suppose that the data function f belongs to C ( , T; H − (Ω)) and the function φ belongs to H (Ω). We assume that the solution (φ k ) ≤k≤K of problem (2.7)-(2.8) satis es the condition (3.3). Then, there exists a positive constant C, depending only on T, m λ , M λ , κ λ and γ, such that the following a posteriori error estimate holds between the solution (φ k ) ≤k≤K of problem (2.7)-(2.8) and the solution (φ k δ ) ≤k≤K of problem (2.20)-(2.21), for all t k , ≤ k ≤ K,
(3.13) Likewise, if we consider ψ δ ∈ X − δ in (2.21), the exactness of the quadrature formula (2.16) permits us to conclude, Then, for any ψ ∈ H (Ω), we have: (3.14) Integrating by parts, we obtain: where [.] is the jump through the point α i . Using equality (3.14), we obtain: The de nition of the discrete product (2.17) yields: And nally, using (2.19) we conclude, Let z δ the image of ψ by a local regularization operator (see [6] for the properties of such operator) where θ i are continuous functions, a ne on each Ω i , equal to in α i and to in the other nodes, (z δ ∈ X δ since ψ ∈ H (Ω)).

Equation (3.16) can be written, by making appear the term
, as follows: Thanks to Cauchy-Schwarz inequality, we have: Then, using lemma 3.1, we obtain: (3.17) Choosing ψ = φ k − φ k δ and using the inequality ab ≤ a + b in (3.17) leads: Using Cauchy-Schwarz inequality, we have: Applying again the inequality ab ≤ a + b , yields Then, using equality (3.7) and the properties of the function λ in (2.2), (2.22) and (3.3), we deduce that, there exists a positive constant C only depending on T, m λ , M λ , κ λ and γ, such that: We conclude (3.13) by doing the sum on k and applying lemma 2.1.
The full a posteriori estimate subject of the following theorem is the result of propositions 3.1 and 3.2 combined together. .

. An upper bound for the error indicators
In this section, we will focus on the upper bound of the error indicators β k and ζ k,i according to the error estimate.

Proposition 3.3.
Assume that the data function f belongs to C ( , T; H − (Ω)) and the function φ belongs to H (Ω). We assume also the solution (φ k ) ≤k≤K of problem (2.7)- (2.8), such that ∂x φ k ∈ L p (Ω), p > , satis es condition (3.3). Then, there exists a positive constant C depending only on T, m λ , M λ , κ λ , |τ| and γ, such that the following estimate holds for the indicator β k , ≤ k ≤ K : Using the de nition of the local norm in (2.12), we conclude: and In order to estimate the term (3.6), (3.7) and integrate between t k− , t k , where C is a positive constant depending only on T, m λ , M λ , κ λ , |τ| and γ. We use (2.13) and (2.14) to evaluate the norm of φτ. Finally, we conclude the estimation (3.18) by inserting all these estimates into (3.19).
For the following, we will be interested to upper bound the error indicator ζ k,i , then we need to introduce the following lemma (see [6] for its proof).

Lemma 3.2.
For any ψ N belongs to the polynomial space P N (Ω), the inverse inequalities hold where C is a positive constant independent of N.
(3.22) Proof 6. Choosing in equality (3.15), ψ δ = and Bounding the terms in the right-hand side follows the same techniques as in the previous proof. Thanks to Cauchy schwarz and the Hölder inequalities ( p + p * = ), we obtain (3.23) Applying the formula (a + b) ≤ a + b , we have Using the two inverse inequalities (3.20) and (3.21) of lemma 3.2, we obtain The last term in the right-hand side of inequality (3.23) is bounded in the same way as in equation (3.8). Then, there exists a positive constant C depending only on T, m λ , M λ , κ λ , |τ| and γ, such that Finally, by inserting (3.24), (3.25) and (3.26) in (3.23), simplifying by and multiplying by N − i , we derive the inequality (3.22).

Conclusion
The a posteriori analysis of the discretization of a partial di erential equations is a very e cient tool for mesh adaptivity. In this paper, we were interested in the a posteriori analysis of the discretization of the non linear heat equation by the spectral element method. We constructed two residual type of indicators and we proved their optimal upper and lower error bounds.