Time and space meshes adaptivity for the resolution of a semi-linear heat equation

Abstract

The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

Keywords

Semi-linear heat equation implicit Euler scheme spectral elements discretization mesh adaptivity error indicators

1. Introduction

The a posteriori analysis of the error estimate and mesh adaptivity for the finite element approximation of partial differential equations has generated considerable interest from researchers and engineers over the past decade (see [3,5,6,15–19]). While few works have been interested in the adaptivity of the discretization by the spectral elements method (see [1,10,11]).

In this work, we are interested on the a posteriori error estimate and the mesh adaptivity in time and in space of semi-linear heat equation modeled by problem (1). We assume that Ω a bounded simply connected domain of $R^{d}$ ( $d = 1, 2$ or 3). Let Γ its Lipschitz continuous boundary and T a positive real number. $\begin{matrix} (1) & \{\begin{matrix} \frac{\partial ω}{\partial t} - div (ζ (ω) \nabla ω) = f & in Ω \times] 0, T [ \\ ω (x, t) = 0 & on Γ \times] 0, T [ \\ ω (x, 0) = ω_{0} & in Ω . \end{matrix} \end{matrix}$ ζ is a continuously differentiable function from $R$ into $R$ satisfying, $\begin{matrix} 0 < m_{ζ} ⩽ ζ (x) ⩽ M_{ζ} and | ζ^{'} (x) | ⩽ κ_{ζ} . \end{matrix}$ The data are the distribution f and the function $ω_{0}$ .

The discretization is based on the backward Euler method in time and on the spectral element method in space. This paper is an extension of the results obtained by Bernardi et al. (see [10]) for a discretization of a linear heat equation by the finite element method and Chorfi et al. (see [2,11]) for the discretization by the spectral element method. For the mesh adaptivity, two local residual error indicators families are defined (see [13]). A first family is related to the time discretization and depends on the semi-discrete solution, the function ζ and the time step. A second family of error indicators is associated to the spectral discretization and depends explicitly on the discrete solution, the function ζ and the data f. The error estimate is upper and lower bounded by the Hilbertian sum of those indicators.

We present some numerical experiments for different meshes in time and in space. Those tests confirm the interest of the spectral element discretization. We show that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal in the sense that the order of convergence is similar to the order of the error indicators.

The outline of the paper is as follows:

In Section 2, we describe the time dicretization using the implicit Euler method and the full discrete problem based on the spectral element discretization.

Section 3 is devoted to the construction of error indicators for the semi-linear heat equation dependant on the discrete solution and the function ζ.

Section 4 deals with some numerical experiments with a focus on the mesh adaptivity in time and in space.

2. The time and the space discretizations

Before studying the variational formulation of the problem (1), we recall the definition of the following Sobolev spaces.

$H^{s} (Ω)$ $s ⩾ 0$ , is the Sobolev space provided with the norm $‖ \cdot ‖_{s, Ω}$ and the semi-norm $| \cdot |_{s, Ω}$ ,

$D (Ω)$ is the space of indefinitely differentiable functions with a compact support in Ω,

$H_{0}^{s} (Ω)$ is the closure of $D (Ω)$ in $H^{s} (Ω)$ and $H^{- s} (Ω)$ its dual space,

the scalar product in the space $L^{2} (Ω)$ is denoted by $(\cdot, \cdot)$ ,

$C^{0} (0, T; H^{s} (Ω))$ is the space of continuous functions, with values in $H^{s} (Ω)$ ,

$L^{2} (0, T; H^{s} (Ω))$ is the space of square-integrable functions with values in $H^{s} (Ω)$ ,

$L^{2} (0, T; H_{0}^{s} (Ω))$ is the space of square-integrable functions with values in $H_{0}^{s} (Ω)$ .

Then, we easily prove that the problem (1) is equivalent to the following variational formulation:

Find $ω \in C^{0} (0, T, L^{2} (Ω)) \cap L^{2} (0, T, H_{0}^{1} (Ω))$ satisfying, $\begin{matrix} (2) & ω (x, 0) = ω_{0} (x), x \in Ω \end{matrix}$ and $\forall t \in] 0, T [$ , $\forall ψ \in H_{0}^{1} (Ω)$ ; $\begin{matrix} (3) & (\frac{\partial ω}{\partial t} (\cdot, t), ψ) + (ζ (ω (\cdot, t)) \nabla ω (\cdot, t), \nabla ψ) = ⟨ f (\cdot, t), ψ ⟩, \end{matrix}$ $⟨ \cdot, \cdot ⟩$ denotes the duality product between the two spaces $H^{- 1} (Ω)$ and $H_{0}^{1} (Ω)$ .

We assume that, if the distribution f belongs to $L^{2} (0, T; L^{2} (Ω))$ and $ω_{0}$ in $L^{2} (Ω)$ , the problem (2)–(3) has a solution $ω \in C^{0} (0, T, L^{2} (Ω)) \cap L^{2} (0, T, H_{0}^{1} (Ω))$ , see (([14], Chap. 7), and ([4,12])).

We consider the energy norm at the time t:

for any $ω \in L^{2} (0, T, H_{0}^{1} (Ω))$ , $\begin{matrix} {[[ω]]}^{2} (t) = ‖ ω ‖_{L^{2} (Ω)}^{2} + m_{ζ} \int_{0}^{t} {‖ \nabla ω (\cdot, s) ‖}_{L^{2} {(Ω)}^{d}}^{2} d s . \end{matrix}$

Then, if we choose the test function $ψ = ω (\cdot, t)$ in (3) and integrating between 0 and t, we prove that the solution ω of problem (2)–(3) satisfies the following stability condition: for $t \in [0, T]$ $\begin{matrix} {[[ω]]}^{2} (t) ⩽ ‖ ω_{0} ‖_{L^{2} (Ω)}^{2} + \frac{1}{m_{ζ}} ‖ f ‖_{L^{2} (0, t; L^{2} (Ω))}^{2} . \end{matrix}$

2.1. The time discretization

For the time discretization, we introduce a partition of the interval $[0, T]$ in sub-intervals $[t_{i}, t_{i + 1}]$ , $1 ⩽ i ⩽ I$ , such that $0 = t_{0} < t_{1} < \dots < t_{I} = T$ . We define the step $h_{i} = t_{i} - t_{i - 1}$ , $h = (h_{1}, \dots, h_{I})$ and $| h | = {max}_{1 ⩽ i ⩽ I} | h_{i} |$ . Let the regularity parameter, $\begin{matrix} μ_{h} = max_{2 ⩽ i ⩽ I} \frac{h_{i}}{h_{i - 1}} . \end{matrix}$ For this partition of the interval $[0, T]$ , we associate a piecewise affine function $ω_{h}$ such that, $\begin{matrix} \forall t \in [t_{i - 1}, t_{i}], ω_{h} (t) = ω^{i} - \frac{t_{i} - t}{h_{i}} (ω^{i} - ω^{i - 1}), \end{matrix}$ where $ω^{i} = ω (\cdot, t_{i})$ .

Using Euler backward method, we deduce from the problem (1) the following time discrete problem: $\begin{matrix} (4) & \{\begin{matrix} \frac{ω^{i} - ω^{i - 1}}{h_{i}} - div (ζ (ω^{i}) \nabla ω^{i}) = f^{i} & in Ω, 1 ⩽ i ⩽ I, \\ ω^{i} = 0 & on Γ, 1 ⩽ i ⩽ I, \\ ω^{0} = ω_{0} & in Ω, \end{matrix} \end{matrix}$ where $f^{i} = f (\cdot, t^{i})$ .

It is easily shown that the problem (4) is equivalent to the following variational formulation:

for $f \in C^{0} (0, T, H^{- 1} (Ω))$ and $ω_{0} \in L^{2} (Ω)$ , find ${(ω^{i})}_{0 ⩽ i ⩽ I} \in L^{2} (Ω) \times {(H_{0}^{1} (Ω))}^{I}$ such that for any $1 ⩽ i ⩽ I$ and $ψ \in H_{0}^{1} (Ω)$ : $\begin{array}{l} (5) & (ω^{i}, ψ) + h_{i} (ζ (ω^{i}) \nabla ω^{i}, \nabla ψ) = (ω^{i - 1}, ψ) + h_{i} ⟨ f^{i}, ψ ⟩ in Ω, \\ (6) & ω^{0} = ω_{0}, in Ω . \end{array}$ Problem (5)–(6) has a unique solution ${(ω^{i})}_{0 ⩽ i ⩽ I} \in L^{2} (Ω) \times {(H_{0}^{1} (Ω))}^{I}$ . This solution satisfies the following stability condition: $\begin{matrix} {‖ ω^{i} ‖}_{L^{2} (Ω)}^{2} + m_{ζ} \sum_{j = 1}^{i} h_{j} {‖ \nabla ω^{j} ‖}_{L^{2} {(Ω)}^{d}}^{2} ⩽ ‖ ω_{0} ‖_{L^{2} (Ω)}^{2} + \frac{1}{m_{ζ}} \sum_{j = 1}^{i} h_{j} {‖ f^{j} ‖}_{H^{- 1} (Ω)}^{2} . \end{matrix}$ See ([2], Section 2) for the proof of the above results.

The time “local” norm is defined by: for any $φ^{i}$ in $H_{0}^{1} (Ω)$ $\begin{array}{rcl} [[φ^{i}]] = {({‖ φ^{i} ‖}_{L^{2} (Ω)}^{2} + h_{i} {| φ^{i} |}_{H^{1} (Ω)}^{2})}^{\frac{1}{2}}, \end{array}$ and the full time discrete norm is defined by: for all ${(φ^{i})}_{0 ⩽ i ⩽ I} \in L^{2} (Ω) \times {(H_{0}^{1} (Ω))}^{I}$ $\begin{matrix} {[[φ^{i}]]}_{i} = {(‖ φ_{0} ‖_{L^{2} (Ω)}^{2} + m_{ζ} \sum_{j = 1}^{i} h_{j} {‖ \nabla φ^{j} ‖}_{L^{2} {(Ω)}^{d}}^{2})}^{\frac{1}{2}} . \end{matrix}$

Then we conclude that for each ${(ω^{i})}_{0 ⩽ i ⩽ I}$ in $L^{2} (Ω) \times {(H^{1} (Ω))}^{I}$ (see [2] for the proof), we have $\begin{matrix} \frac{1}{4} {[[ω^{i}]]}_{i}^{2} ⩽ {[[ω_{h}]]}^{2} (t_{i}) ⩽ \frac{1}{2} (1 + μ_{h}) {[[ω^{i}]]}_{i}^{2} + \frac{m_{ζ}}{2} h_{1} ‖ \nabla ω_{0} ‖_{L^{2} {(Ω)}^{d}}^{2} . \end{matrix}$

2.2. The spectral element discretization

In this section, we handle only the one dimension case since for the spectral method the inverse inequalities are not optimal in dimension $d ⩾ 2$ . We assume $Ω =] - 1, 1 [$ , and its spectral elements decomposition is: $\begin{matrix} Ω = ⋃_{k = 1}^{K} Ω_{k}, \end{matrix}$ where $Ω_{k} =] b_{k - 1}, b_{k} [$ such that $- 1 = b_{0} < b_{1} < \dots < b_{k} < \dots < b_{K} = 1$ for $1 ⩽ k ⩽ K$ .

Let the step $τ_{k} = (b_{k} - b_{k - 1})$ and an integer $N_{k} ⩾ 2$ associated to the sub-domain $Ω_{k}$ , we define the discrete parameter: $\begin{matrix} δ = ((τ_{1}, N_{1}), (τ_{2}, N_{2}), \dots, (τ_{K}, N_{K})) . \end{matrix}$ We consider $P_{N} (Ω)$ the space of polynomials of degree $N ⩾ 2$ . To define the spectral discrete problem, we need to recall the following Gauss-Lobatto quadrature formula: $\begin{matrix} (7) & \forall φ \in P_{2 N - 1} (Ω); \int_{- 1}^{1} φ (x) d x = \sum_{n = 0}^{N} φ (ϵ_{n}^{N}) ρ_{n}^{N} \end{matrix}$ where the nodes $ϵ_{0} < ϵ_{1} < \dots < ϵ_{N}$ are the zeros in Ω of the polynomial

$(1 - x^{2}) L_{N}^{'} (x)$ . $L_{N}$ is the Legendre polynomial defined on Ω, and $ρ_{n}$ , $0 ⩽ n ⩽ N$ , represent the associated positive weights.

For a continuous functions u and v on $\overline{Ω}$ , the discrete scalar product is defined by: $\begin{matrix} {(u, v)}_{δ} = \sum_{k = 1}^{K} \sum_{n = 0}^{N_{k}} u (ϵ_{n}^{N_{k}}) v (ϵ_{n}^{N_{k}}) ρ_{n}^{N_{k}}, \end{matrix}$ such that $ϵ_{n}^{N_{k}} = T_{k}^{- 1} (ϵ_{n})$ the local nodes and $ρ_{n}^{N_{k}} = (b_{k} - b_{k - 1}) ρ_{n}$ the local weights, $0 ⩽ n ⩽ N$ , where $T_{k}$ is the bijection from $Ω_{k}$ into Ω.

We define the space: $\begin{matrix} Y_{δ} = {u_{δ} \in H^{1} (Ω); u_{δ} /_{Ω_{k}} \in P_{N_{k}} (Ω_{k}), 1 ⩽ k ⩽ K}, \end{matrix}$ then, we recall the following important property (see [9]), $\begin{matrix} (8) & \forall u_{δ} \in Y_{δ}, ‖ u_{δ} ‖_{L^{2} (Ω)}^{2} ⩽ {(u_{δ}, u_{δ})}_{δ} ⩽ 3 ‖ u_{δ} ‖_{L^{2} (Ω)}^{2} . \end{matrix}$ The Lagrange interpolation operator $π_{δ}$ is defined such that for u continuous on each ${\bar{Ω}}_{k}$ , for $0 ⩽ k ⩽ K$ : $\begin{matrix} π_{δ} (u) /_{Ω_{k}} \in P_{N_{k}} (Ω_{k}) π_{δ} (u) /_{Ω_{k}} (ϵ_{n}^{N_{k}}) = u /_{Ω_{k}} (ϵ_{n}^{N_{k}}) . \end{matrix}$ We consider the discrete space $X_{δ} = Y_{δ} \cap H_{0}^{1} (Ω)$ . Then, our full discrete problem is: Find $ω_{δ}^{i}$ in $Y_{δ} \times Π_{i = 1}^{I} X_{δ}$ such that, $\begin{array}{l} (9) & ω_{δ}^{0} = π_{δ} (ω_{0}) in Ω, \\ (10) & \forall ψ_{δ} \in X_{δ}, {(ω_{δ}^{i}, ψ_{δ})}_{δ} + h_{i} {(ζ_{δ} (ω_{δ}^{i}) \nabla ω_{δ}^{i}, \nabla ψ_{δ})}_{δ} = {(ω_{δ}^{i - 1}, ψ_{δ})}_{δ} + h_{i} {(f^{i}, ψ_{δ})}_{δ} . \end{array}$ For each function ω continuous on $\bar{Ω}$ , $ζ_{δ}$ is defined by: $\begin{matrix} ζ_{δ} (ω) = min {max {π_{δ} (ζ (ω)); m_{ζ}}; M_{ζ}} . \end{matrix}$ Based on the Brouwer’s fixed point theorem and the inequality (8), we obtain that for $f^{i} \in H^{- 1} (Ω)$ $1 ⩽ i ⩽ I$ , $ω^{0} \in L^{2} (Ω)$ and $ω_{δ}^{i - 1} \in X_{δ}$ , the problem (9)–(10) has a unique solution $ω_{δ}^{i}$ in $X_{δ}$ . This solution satisfies: $\begin{matrix} {[[ω_{δ}^{i}]]}_{i}^{2} ⩽ ‖ π_{δ} ω_{0} ‖_{L^{2} (Ω)}^{2} + \frac{1}{m_{ζ}} \sum_{j = 1}^{i} h_{j} {‖ f^{j} ‖}_{H^{- 1} (Ω)}^{2} . \end{matrix}$

3. Error indicators and adaptivity

This section deals with the definition of the two families of error indicators. The first indicator is related to time discretization and the second one to spectral element discretization [7].

The time error indicators are defined as: for each $1 ⩽ i ⩽ I$ , $\begin{array}{rcl} η_{i} = {(\frac{h_{i}}{3})}^{\frac{1}{2}} {‖ ζ_{δ} {(ω_{δ}^{i})}^{\frac{1}{2}} \frac{d}{d x} (ω_{δ}^{i} - ω_{δ}^{i - 1}) ‖}_{L^{2} (Ω)} . \end{array}$

We also define the space local indicators which depend of the discrete solution:

for each i, $1 ⩽ i ⩽ I$ and each sub-interval $Ω_{k}$ , $1 ⩽ k ⩽ K$ $\begin{array}{rcl} ϖ_{i, k} = N_{k}^{- 1} {‖ (π_{δ} f^{i} - \frac{ω_{δ}^{i} - ω_{δ}^{i - 1}}{h_{i}} + \frac{d}{d x} (ζ_{δ} (ω_{δ}^{i}) \frac{d ω_{δ}^{i}}{d x})) {(x - b_{k - 1})}^{\frac{1}{2}} {(b_{k} - x)}^{\frac{1}{2}} ‖}_{L^{2} (Ω_{k})} . \end{array}$ We use the following triangular inequality to prove that the error estimate is equivalent to the error indicators: $\begin{matrix} [[ω - ω_{δ}]] (t_{i}) ⩽ [[ω - ω_{h}]] (t_{i}) + [[ω_{h} - ω_{δ}]] (t_{i}) . \end{matrix}$ We define $π_{h}$ the time interpolating operator as: for a continuous function u on the interval $[0, T]$ , $π_{h} u$ is equal to $u (t_{i})$ on each sub-interval $[t_{i - 1}, t_{i}]$ , $1 ⩽ i ⩽ I$ . We estimate in the following proposition the term $[[ω - ω_{h}]] (t_{i})$ where ω is the solution of problem (1), and $ω_{δ}$ is the solution of problem (9)–(10), (see [2] for its proof).

Proposition 3.1.
We assume that the data function f belongs to $C^{0} (0, T; H^{- 1} (Ω))$ , the function $ω_{0}$ belongs to $H_{0}^{1} (Ω)$ and the solution ${(ω^{i})}_{0 ⩽ i ⩽ I}$ of problem ( 4 ) satisfies, for $r > 1$ , $\begin{matrix} (11) & sup_{0 ⩽ i ⩽ I} {‖ \frac{d ω^{i}}{d x} ‖}_{L^{r} (Ω)} ⩽ β, \end{matrix}$ where β is a constant depending only on the data f, $ω_{0}$ and ζ. Then, there exists a positive constant C, depending only on T, $m_{ζ}$ , $M_{ζ}$ , $κ_{ζ}$ and β, such that the following bound of the error estimate holds $\begin{array}{rcl} [[ω - ω_{h}]] (t_{i}) & ⩽ & C ({(1 + μ_{h})}^{\frac{1}{2}} [[ω_{h} - ω_{δ}]] (t_{i}) + {(\sum_{j = 1}^{i} η_{j}^{2})}^{\frac{1}{2}} \\ + ‖ f - π_{h} f ‖_{L^{2} (0, t_{i}; H^{- 1} (Ω))}) . \end{array}$

For the estimation of the term $[[ω_{h} - ω_{δ}]] (t_{i})$ , $1 ⩽ i ⩽ I$ presented in the following proposition (see [2] for the proof), we use the next result: for any $u \in H^{r} (] - 1, 1 [) \cap H_{0}^{1} (] - 1, 1 [)$ , $r > 1$ , $\begin{array}{rcl} {(\int_{- 1}^{1} {(u - Π_{N}^{1, 0} u)}^{2} (x) {(1 - x^{2})}^{- 1} d x)}^{\frac{1}{2}} ⩽ C N^{- r} ‖ u ‖_{H^{r} (Ω)}, \end{array}$ where $Π_{N}^{1, 0}$ is the orthogonal projection operator defined from $H_{0}^{1} (Ω)$ into $P_{N} (Ω) \cap H_{0}^{1} (Ω)$ (See [8,9]).
Proposition 3.2.
We assume that the data function f belongs to $C^{0} (0, T; H^{- 1} (Ω))$ the function $ω_{0}$ belongs to $H_{0}^{1} (Ω)$ and the solution ${(ω^{i})}_{0 ⩽ i ⩽ I}$ of problem ( 5 )–( 6 ) satisfies the condition ( 11 ). Then, there exists a positive constant C, depending only on T, $m_{ζ}$ , $M_{ζ}$ , $κ_{ζ}$ and β, such that, $\begin{matrix} [[ω_{h} - ω_{δ}]] (t_{i}) ⩽ & C {(\sum_{j = 1}^{i} h_{j} \sum_{k = 1}^{K} (ϖ_{j, k}^{2} + {‖ f^{j} - π_{δ} f^{j} ‖}_{L^{2} (Ω_{k})}^{2}))}^{\frac{1}{2}} \\ + ‖ ω_{0} - π_{δ} ω_{0} ‖_{L^{2} (Ω)} . \end{matrix}$

Now, if we combine the estimations in propositions (3.1) and (3.2), we conclude the full a posteriori estimate in the following theorem:
Theorem 3.1.
We assume that the data function f belongs to $C^{0} (0, T; H^{- 1} (Ω))$ , the function $ω_{0}$ belongs to $H_{0}^{1} (Ω)$ and the solution ${(ω^{i})}_{0 ⩽ i ⩽ I}$ of problem ( 5 )–( 6 ) satisfies the condition ( 11 ). There exists a positive constant C depending only on T, $m_{ζ}$ , $M_{ζ}$ , $κ_{ζ}$ and β, such that the full a posteriori estimate holds. $\begin{matrix} [[ω - ω_{δ}]] (t_{i}) ⩽ & C {(\sum_{j = 1}^{i} (η_{j}^{2} + h_{j} \sum_{k = 1}^{K} (ϖ_{j, k}^{2} + {‖ f^{j} - π_{δ} f^{j} ‖}_{L^{2} (Ω_{k})}^{2})))}^{\frac{1}{2}} \\ + (‖ ω_{0} - π_{δ} ω_{0} ‖_{L^{2} (Ω)} + ‖ f - π_{h} f ‖_{L^{2} (0, t_{i}, H^{- 1} (Ω))}) . \end{matrix}$

For the time and spectral mesches adaptivity, the bound of the indicators $η_{i}$ and $ϖ_{i, k}$ by the error estimate is requiered. We announce these results in the two following propositions proved in [2], Proposition 3.3.
We suppose that the data f belongs to $C^{0} (0, T; H^{- 1} (Ω))$ , the function $ω_{0}$ belongs to $H_{0}^{1} (Ω)$ and the solution ${(ω^{i})}_{0 ⩽ i ⩽ I}$ of problem ( 5 )–( 6 ) satisfies the condition ( 11 ). There exists a positive constant C depending only on T, $m_{ζ}$ , $M_{ζ}$ , $κ_{ζ}$ , $| h |$ and β, such that the estimate for the time indicator $η_{i}$ , $1 ⩽ i ⩽ I$ , holds: $\begin{matrix} η_{i} ⩽ & C ([[ω^{i} - ω_{δ}^{i}]] + {(μ_{h})}^{\frac{1}{2}} [[ω^{i - 1} - ω_{δ}^{i - 1}]] + {(h_{i})}^{\frac{1}{2}} ‖ f - π_{h} f ‖_{L^{2} (t_{i - 1}, t_{i}; H^{- 1} (Ω))} \\ + {(h_{i})}^{\frac{1}{2}} ({‖ \partial_{t} (ω - ω_{h}) ‖}_{L^{2} (t_{i - 1}, t_{i}; H^{- 1} (Ω))} \\ + {‖ \partial_{x} (ω - ω_{h}) ‖}_{L^{2} (t_{i - 1}, t_{i}; L^{2} (Ω))})) . \end{matrix}$
Proposition 3.4.
We suppose that the data f belongs to space $C^{0} (0, T; H^{- 1} (Ω))$ the function $ω_{0}$ belongs to $H_{0}^{1} (Ω)$ and the solution ${(ω^{i})}_{0 ⩽ i ⩽ I}$ of problem ( 5 )–( 6 ) satisfies the condition ( 11 ). There exists a positive constant C depending only on T, $m_{ζ}$ , $M_{ζ}$ , $κ_{ζ}$ , $| h |$ and β. The estimate for the spectral indicator $ϖ_{i, k}$ , $1 ⩽ k ⩽ K$ , $1 ⩽ i ⩽ I$ holds, $\begin{matrix} ϖ_{i, k} ⩽ & C ({‖ \partial_{x} (ω^{i} - ω_{δ}^{i}) ‖}_{L^{2} (Ω_{k})} + {‖ \frac{(ω^{i} - ω_{δ}^{i}) - (ω^{i - 1} - ω_{δ}^{i - 1})}{h_{i}} ‖}_{H^{- 1} (Ω_{k})} \\ + N_{k}^{- 1} τ_{k} {‖ f^{i} - π_{δ} f^{i} ‖}_{L^{2} (Ω_{k})}) . \end{matrix}$

4. Numerical results

In this section, we present some numerical tests in order to illustrate the adaptivity of meshes in time and in spectral discretization. we show by some curves the equivalence between the error time indicators and the space error indicators with the error estimate.

The computing of the nonlinear term ${(ζ_{δ} (ω_{δ}^{i}) \nabla ω_{δ}^{i}, \nabla ψ_{δ})}_{δ}$ in the discrete problem (10) by the quadrature formula (7) is made simpler by choosing a simple value of ζ such that $ζ_{δ} (φ_{δ}^{i}) \nabla φ_{δ}^{i} \nabla ψ_{δ}$ is a polynomial of degree $⩽ 2 N_{k} - 1$ on each sub-domain $Ω_{k}$ .

We define the hilbertian sum of the local spectral indicators $ϖ_{i, k}$ as: $\begin{matrix} ϖ^{i} = \sum_{k = 1}^{k = K} ϖ_{i, k} . \end{matrix}$ We consider the domain $Ω =] - 1, 1 [$ . We begin by the first test where the solution ω of the problem (1) is regular and the function ς represents the linear case. $\begin{matrix} (12) & ω (x, t) = e^{t} sin (π x), ς (x) = 1 . \end{matrix}$ Second, we handle the case where the solution ω of the problem (1) is singular and the function ς is a polynomial function. $\begin{matrix} (13) & ω (x, t) = t^{\frac{1}{2}} {(1 - x^{2})}^{\frac{7}{2}}, ς (x) = x . \end{matrix}$ We fix the spectral discrete parameter $δ = ((\frac{1}{2}, 20), (\frac{1}{2}, 20))$ , $T = 1$ and the time steps h vary in the set ${0.1, 0.01, 0.001, 0.0001, 0.00001, 0.000001}$ .

Fig. 1.

The time adaptivity.

Figure 1 presents the curves of convergence for the two terms $log ‖ Error ‖_{L^{2} (Ω)}$ (in blue), and $log (η_{i})$ (in red) as a function of $log (h)$ . Figure 1(a) respectively Fig. 1(b) represents the convergence in time for the continuous solutions defined in (12) respectively for the solution defined in (13). We remark that, in the two cases (regular solution or less regular solution), the time convergence order is almost equal to 1 and the equivalence between the error $Error$ and the time error indicator $η^{i}$ when the spectral mesh is fixed. This confirms the results of the Proposition 3.1 and Proposition 3.3.

The next test is about the adaptivity when the time mesh is fixed for step $h = 0.001$ and $T = 1$ while the spectral element mesh δ is within $((\frac{1}{5}, N), (\frac{1}{5}, N), (\frac{1}{5}, N), (\frac{1}{5}, N), (\frac{1}{5}, N))$ with N varying in the set ${5, 7, 15, 25, 35, 40}$ .

Fig. 2.

The spectral discretization adaptivity.

Figure 2 shows the convergence curves of the relative errors $Error$ in norm $L^{2} (Ω)$ in (blue), and $log (ϖ^{i})$ (in red) in semi-logarithmic scales, as a function of $log (N)$ . In Fig. 2(a), the convergence is about the regular solution defined in (12). The accuracy is much better than the convergence in Fig. 2(b) of the singular solution defined in (13). We note that, in the two cases (regular solution or less regular solution), the error $Error$ is equivalent to the spectral indicator $ϖ^{i}$ when the time mesh is fixed. this confirms the results of the proposition (3.2) and proposition (3.4).

In the last test, we study the influence of meshes related to the time and space on the time indicators $η^{i}$ and the spectral indicators $ϖ^{i}$ .

Fig. 3.

Correlation between the error, $η^{i}$ and $ϖ^{i}$ .

We present in Fig. 3(a) the error $Error$ (in blue), the error time indicator $η^{i}$ (in green) and the spectral error indicator $ϖ^{i}$ (in red). When the parameter spectral discretization N is fixed, we remak that the spectral error indicator $ϖ^{i}$ does not depend on time while the time error indicator $η^{i}$ and the error $Error$ decrease with exactly the same slope until the error becomes larger than the spectral indicator $ϖ^{i}$ .

We present in Fig. 3(b) the error $Error$ (in blue), the error time indicator $η^{i}$ (in green) and the spectral error indicator $ϖ^{i}$ (in red). When the time step h is fixed, we note that the time error indicators $η^{i}$ are completely independent of N. Moreover, the spectral error indicator $ϖ^{i}$ and the error $Error$ decrease with exactly the same slope until the error becomes larger than the time indicator $η^{i}$ .

Footnotes

Acknowledgements

This project was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, award number 14-MAT739-02.

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