Concluding Remarks

CHAPTER 4. Semidiscrete error analysis by energy arguments 77 term as in Example 4.1) with mesh size h= 1/20,1/40,1/80,1/160,1/320 with a fixed time step size k = 0.00002. Since the nonsmooth initial data error estimate reflects a singularity at t = 0 (see Theorem 4.3.2), we particularly interested in the behavior of the errors when T is close to zero. For this purpose, we choose T = 0.001 in our computation. The empirical CRs of the errors in both L²(Ω)- and H¹(Ω)-norms are listed in Table 4.3 which validate the theoretical CRs of Theorem 4.3.2.

Table 4.3: The L²(Ω)- and H¹(Ω)-errors and CRs for Example 4.3 at T = 0.001 with α = 0.35,0.55,0.75 and k= 0.00002

hα= 0.35 L²-norm H¹-norm L²-CR H¹-CR

1/40 1.19900e-03 4.48736e-02 − −

1/80 3.16727e-04 2.33723e-02 1.92052 0.94106 1/160 8.10442e-05 1.16394e-02 1.96646 1.00578 1/320 2.03453e-05 5.77652e-03 1.99401 1.01075

α= 0.55

2.52952e-03 9.68415e-02 − − 6.85072e-04 5.08080e-02 1.88453 0.93057 1.77321e-04 2.53730e-02 1.94989 1.00176 4.47661e-05 1.25941e-02 1.98588 1.01055

α= 0.75

5.65607e-03 2.16012e-01 − − 1.57299e-03 1.14010e-01 1.84629 0.92194 4.08456e-04 5.67961e-02 1.94526 1.00530 1.03102e-04 2.81395e-02 1.98610 1.01320

CHAPTER 4. Semidiscrete error analysis by energy arguments 78 gence order of the approximate solution for bothL²(Ω)- andH¹(Ω)-norms. Further, we have established a quasi-optimal pointwise in time error bound in the L^∞(Ω)-norm for smooth initial data. Extensive numerical illustrations confirm our theoretical conver- gence results.

5

Fully Discrete Schemes

This chapter studies the error estimates for the fully discrete finite element approxima- tions for problem (3.1) withB(t, s) = −A. We consider two fully discrete schemes based on the convolution quadrature, namely, the backward Euler (BE) and the second-order backward difference (SBD). The convergence analysis is analyzed for both smooth and nonsmooth data cases and optimal error bounds in time are derived in theL^∞(L²)-norm for both schemes.

5.1 Introduction

We first recall the SEM with B(t, s) = −A, i.e.,

(5.1)

∂_t^αu(t) +Au(t) + Z t

0

Au(s)ds =f(t) in Ω, 0< t≤T, 0< α <1, u(t) = 0 on ∂Ω, 0< t≤T,

u(0) =u₀(x) in Ω.

We now briefly discuss the convolution quadrature. This numerical method was first introduced by Lubich in [38, 39] and subsequently used by many authors in [5, 11, 27, 28, 40, 41, 61, 64]. The main advantage of using the convolution quadrature is that it allows the discretization of the derivative and the integral term in (5.1) simultaneously.

However, this is not the case for the standard temporal discretization where the deriva- tive and the integral terms are discretized separately. This procedure requires a higher temporal regularity of the solution uin the error analysis which problem (5.1) does not possess even for smooth initial data (see Theorems 2.4.1, 2.4.2, and 2.4.3). The con- volution quadrature framework provides excellent stability properties and higher order accuracy in time. It also permits omission of the term corresponding to t = 0 during discretization keeping the convergence rate unchanged.

CHAPTER 5. Fully discretization 80 Our study to (5.1) is motivated by the works of Bazhlekova et al. [5] and Jin et al. [27]. In [5], the authors have studied the Rayleigh-Stokes problem for a generalized second-grade fluid. Both semidiscrete and fully discrete schemes are discussed and optimal order error bounds are established covering both smooth and nonsmooth initial data cases. The semidiscrete scheme is based on the piecewise linear GFEM in space and the fully discrete scheme is based on the convolution quadrature in time with the generating functions given by the BE and SBD methods. In [27], the authors have considered and analyzed the fully discrete BE and SBD schemes for the SE and derived error estimates of first-order and second-order accuracy in time, respectively, for both smooth and nonsmooth initial data. In this chapter, we have made an attempt to extend these results to SEM.

Next, we describe the convolution quadrature framework. Let K be a complex- valued or operator valued analytic function in the sector S_π−ϑ = {z ∈ C : |arg z| ≤ ϑ},0< ϑ < π/2 and satisfies

(5.2) kK(z)k ≤M₀|z|^−γ,

for some real numbersM0, γ and for all z ∈Sπ−ϑ. Then there exists a distributionκ(t) on the real line R such that K(z) = L{κ(t)}, the Laplace transform of κ(t). Here the distribution κ(t) for positive time is given by the following inversion formula

κ(t) = 1 2πi

Z

Γ

K(z)e^ztdz,

where Γ is a contour in the sectorS_π−ϑ, parallel to its boundary and with imaginary part increasing. Further, κ(t) is zero on the negative real axis and it has singular support empty or concentrated at the origin. Let us defineK(D_t) withD_t = _dt^d as the operator of (distributional) convolution with the kernelκ: K(D_t)f :=κ∗f =Rt

0 κ(t−τ)f(τ)dτ, t >

0, for a suitable function f(t). Let K₁ and K₂ be two operators of type K generated by the kernels κ₁ and κ₂, respectively. Then we have the following relation using the convolution rule

(5.3) K1(Dt)K2(Dt)f(t) = (K1K2)(Dt)f(t).

Now consider a partition of [0, T] with equidistant points{t_n}^N_n=0, wheret_n =nk, n= 0, . . . , N with k = T /N. Let U_hⁿ be the approximation of u(t_n). The convolution quadrature of the continuous convolution K(D_t)f(t) is denoted by K( ¯D_k)f(t) and is defined by (cf. [11, 39])

(5.4) K( ¯Dk)f(t) = X

0≤jk≤t

ωjf(t−jk), t >0,

CHAPTER 5. Fully discretization 81 where the quadrature weights ωj, j = 0,1, . . . are determined byP∞

j=0ωjξ^j =K(^δ(ξ)_k ),δ is the quotient of the generating polynomials of a stable and consistent linear multistep method. For the BE methodδ(ξ) = (1−ξ) and for the SBD methodδ(ξ) = ³₂−2ξ+^ξ₂². Further, similar to (5.3) the convolution quadrature satisfies the following relations:

(5.5) K₁( ¯D_k)K₂( ¯D_k)f(t) = (K₁K₂)( ¯D_k)f(t), and K₁( ¯D_k)(κ₂∗f) = (K₁( ¯D_k)κ₂)∗f.

We now proceed to estimate the errors in the fully discrete approximations. This is accomplished by reformulating the problem by transforming the Caputo fractional derivative to the Riemann-Liouville, and then apply the convolution quadrature. It is known fact that the Caputo and the Riemann-Liouville fractional time derivatives (cf.

[30, p.91] or [27, p.A150]) are related as follows: For ˜α∈(m−1, m), m∈N, (5.6) ∂_t^α˜f(t) = D_t^α˜

f(t)−

m−1

X

j=0

f^j(0) j! t^j

,

where D_t^α˜f(t) is the Riemann-Liouville fractional time derivative (left-sided) of order ˜α [55], defined by

(5.7) D^α_t^˜f(t) = 1

Γ(m−α)˜ d^m dt^m

Z t 0

f(τ)

(t−τ)^{1+ ˜α−m} dτ.

In our analysis, we considerm= 1 in (5.6) and hence, ∂_t^α˜f(t) = D_t^α˜(f(t)−f(0)). Using this identity and B_h(t, s) = −A_h we can rewrite the semidiscrete approximation (3.4) as

(5.8) u_h(t) =

D_t^α+ (I+D⁻¹_t )A_h−1

D_t^αu_0h+

D_t^α+ (I+D_t⁻¹)A_h−1

f_h(t), t >0.

As usual, the smooth data error analysis is carried out for the nonhomogeneous problem whereas the homogeneous problem is considered for nonsmooth data error estimate.

The outline of this chapter is as follows. We study the BE and SBD schemes, respec- tively, in Sections 5.2 and 5.3. In Section 5.4, we present some numerical illustrations to validate our theoretical findings.