A posteriori error estimation for quasilinear singularly perturbed problems with integral boundary condition

Abstract

We consider a quasilinear singularly perturbed parametrized problem with integral boundary condition. To solve the problem numerically, the discretization comprises of an implicit Euler scheme for the quasilinear problem and a composite right rectangle rule for the integral boundary condition. We establish a posteriori error estimate for the discrete problem that holds true uniformly in the small perturbation parameter. Further, we rectify the shortcomings of a posteriori error estimation in L.-B. Liu et al. (Numerical Algorithms 83:719–739, 2019) for a different class of problems. Numerical experiments are performed and results are reported for validation of the theoretical error estimates.

Appendix.: Proof of Theorem 3.1

In this appendix we provide the technical results to prove Theorem 3.1. We first give a general error estimate for scheme (3.5) based on which a number of a priori meshes for problem (1.1) can be constructed that can resolve the layer and produce parameter-uniform numerical approximation for problem (1.1). The following bound on the derivative of u proved in [19, Lemma 2.1] is required:

$$ \begin{array}{@{}rcl@{}} |{u^{\prime}(t)}| & \leq C\left\{1+\frac{1}{\varepsilon}e^{-\frac{\beta t}{\varepsilon}}\right\},~t\in\overline{J}. \end{array} $$

(A.1)

Further, similar to the continuous case, we have the following lemma.

Lemma A.1

For any mesh functions (Y,η^N) and (Z,ν^N) such that Y_N = Z_N,

$$ (Y_{0}-Z_{0})+\sum\limits_{j=1}^{N}h_{j}c_{j}(Y_{j}-Z_{j})=\phi, $$

and

$$ \mathcal{T}^{N}_{\eta^{N}}Y-\mathcal{T}_{\nu^{N}}^{N}Z=\chi, $$

we have

$$ \max\{\|{Y-Z}\|_{\overline{\omega}}, |{\eta^{N}-\nu^{N}}|\}\leq C(\|{\chi}\|_{{\omega}}+|{\phi}|). $$

Proof

This proof can be readily done using the argument similar to Lemma 2.2 and using Lemma 3.1. □

We shall use the following notation

$$ \begin{array}{@{}rcl@{}} \vartheta(\omega) & := \underset{1\leq j\leq N}{\max }{\int}_{t_{j-1}}^{t_{j}}\left( 1+\varepsilon^{-1}e^{-\frac{\beta t}{\varepsilon}}\right)dt. \end{array} $$

(A.2)

Theorem A.1

Let {u,λ} be the solution of (1.1) and {U,λ^N} be its approximation using scheme (3.5). Then

$$ \begin{array}{@{}rcl@{}} \max\bigg\{\|{U-u}\|_{\bar{\omega}}, |{\lambda^{N}-\lambda}|\bigg\} & \leq C\vartheta(\omega). \end{array} $$

(A.3)

Proof

Using (3.1), (3.3), and (3.5), we get

$$ \mathcal{T}^{N}_{\lambda^{N}}U-\mathcal{T}_{\lambda}^{N}u=R_{j}~~\text{and}~~(U_{0}-u_{0})+\sum\limits_{j=1}^{N}h_{j}c_{j}(U_{j}-u_{j})=r. $$

Now, for t_j ∈ ω,

$$ \begin{array}{@{}rcl@{}} |{R_{j}}| & \leq& \frac{1}{h_{j}}{\int}_{t_{j-1}}^{t_{j}}(t-t_{j-1})|{f_{t}(t, u(t), \lambda)+f_{u}(t, u(t), \lambda)u^{\prime}(t)}|dt\\ & \leq& \frac{C}{h_{j}}{\int}_{t_{j-1}}^{t_{j}}(t-t_{j-1})(1+|{u^{\prime}(t)}|)dt\\ & \leq& C{\int}_{t_{j-1}}^{t_{j}}(1+|{u^{\prime}(t)}|)dt\\ & \leq& C{\int}_{t_{j-1}}^{t_{j}}(1+\varepsilon^{-1}e^{-\frac{\beta t}{\varepsilon}})dt. \end{array} $$

Also,

$$ \begin{array}{@{}rcl@{}} |{r}| & \leq& {\sum}_{j=1}^{N}{\int}_{t_{j-1}}^{t_{j}}c(t)|{t-t_{j-1}}||{u^{\prime}(t)}vdt\\ & \leq& \|{c}\|C{\sum}_{j=1}^{N}h_{j}{\int}_{t_{j-1}}^{t_{j}}(1+\varepsilon^{-1}e^{-\frac{\beta t}{\varepsilon}})dt\\ & \leq& C\vartheta(\omega){\sum}_{j=1}^{N}h_{j}\\ & \leq& C\vartheta(\omega). \end{array} $$

Hence, using Lemma A.1, we have the desired result. □

We can use Theorem A.1 to construct a number of a priori meshes for problem (1.1). We here describe standard Shishkin and Bakhvalov meshes.

Bakhvalov meshes

There are several ways to construct these meshes. Our construction is based on the equidistribution of the function

$$ \begin{array}{@{}rcl@{}} M_{B}(s) & :=\max\left\{1, \frac{K}{\varepsilon}e^{-\frac{\beta s}{\varepsilon}}\right\} \end{array} $$

(A.4)

where K is a positive user chosen constant; that is, the mesh points t_j are so that

$$ {\int}_{0}^{t_{j}}M_{B}(s)ds=\frac{j}{N}{{\int}_{0}^{1}}M_{B}(s) ds. $$

On this mesh, we have 𝜗(ω) ≤ CN^− 1, cf. [23]. Hence, by Theorem A.1, we get

$$ \max\bigg\{\|{U-u}\|_{\bar{\omega}}, |{\lambda^{N}-\lambda}|\bigg\}\leq CN^{-1}. $$

Shishkin meshes

Defining the transition point τ_s by

$$ \begin{array}{@{}rcl@{}} \tau_{s} &=\min\left\{\frac{1}{2}, \beta^{-1}\varepsilon\ln N\right\}, \end{array} $$

(A.5)

and placing N/2 subintervals on each [0,τ_s] and [τ_s, 1], Shishkin meshes are constructed. The mesh points are defined by

$$ t_{j}=\begin{cases} jh^{(1)},\qquad\qquad\qquad j=0,\cdots, N/2,\\ \tau_{s}+(j-N/2)h^{(2)},\quad j=N/2+1, \cdots, N, \end{cases} $$

(A.6)

where h⁽¹⁾ = 2τ_s/N and h⁽²⁾ = 2(1 − τ_s)/N.

On this mesh, we have $\vartheta (\omega )\leq C(N^{-1}\ln N),$ cf. [23]. Hence, by Theorem A.1, we get

$$ \max\bigg\{\|{U-u}\|_{\bar{\omega}}, |{\lambda^{N}-\lambda}|\bigg\}\leq CN^{-1}\ln N. $$