Numerical Approximation of Space-Time Fractional Parabolic Equations

Andrea Bonito; Wenyu Lei; Joseph E. Pasciak

doi:10.1515/cmam-2017-0032

Published by De Gruyter August 26, 2017

Numerical Approximation of Space-Time Fractional Parabolic Equations

Andrea Bonito , Wenyu Lei and Joseph E. Pasciak

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2017-0032

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Abstract

In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e. an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator E⁢(t) for the initial value problem can be written as a Dunford–Taylor integral involving the Mittag-Leffler function eα,1 and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here α denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator W⁢(t) and the forcing function F⁢(t). We develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford–Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of 1h. The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying a sinc quadrature technique to approximate the Dunford–Taylor integral of the discrete operator and is free of any time stepping. The sinc quadrature of step size k involves k-2 nodes and results in an additional O⁢(exp⁡(-ck)) error. To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo-midpoint quadrature. This involves the average of Wh⁢(s), (the semi-discrete approximation to W⁢(s)) over the quadrature interval. This average can also be written as a Dunford–Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford–Taylor integrals for computing the averages. We show that for a refined grid in time with a mesh of O⁢(𝒩⁢log⁡(𝒩)) intervals, the error between the semi-discrete and fully discrete approximation is O⁢(𝒩-2+log⁡(𝒩)⁢exp⁡(-ck)). We also report the results of numerical experiments that are in agreement with the theoretical error estimates.

Keywords: Time Dependent Problem; Fractional Diffusion; Caputo Fractional Derivative; Finite Element Method; Sinc Quadrature; Dunford–Taylor

MSC 2010: 65M12; 65M15; 65M60; 35S11; 65R20; 34A08; 34K37

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1254618

Funding statement: The first and second authors were partially supported by the National Science Foundation through Grant DMS-1254618.

A Proof of Lemma 3.2

The following lemma proved in [3] (see [3, Lemma 3.1]) and is instrumental in the proof of Lemma 3.2.

Lemma A.1.

There is a positive constant C only depending on s∈[0,1] such that

(A.1)|z|-s⁢∥T1-s⁢(z-1⁢I-T)-1⁢f∥≤C⁢∥f∥ for all ⁢z∈𝒞,f∈L2.

The same inequality holds Vh, i.e. with T replaced by Th and f∈Vh.

Proof of Lemma 3.2.

Noting that

Rz⁢(L)=(z⁢I-L)-1=T⁢(z⁢T-I)-1

and

Rz⁢(Lh)⁢πh=(z⁢I-Lh)-1⁢πh=(z⁢Th-I)-1⁢Th⁢πh=(z⁢Th-I)-1⁢Th,

we obtain

πh⁢Rz⁢(L)-Rz⁢(Lh)⁢πh=πh⁢(T⁢(z⁢T-I)-1-(z⁢Th-I)-1⁢Th)

=πh⁢(z⁢Th-I)-1⁢(Th-T)⁢(z⁢T-I)-1

=-z-2⁢(Th-z-1)-1⁢πh⁢(T-Th)⁢(T-z-1)-1,

where for the last step we used the definition of Th to deduce that πh⁢(z⁢Th-I)-1=(z⁢Th-I)-1⁢πh. We are left to prove

(A.2)∥W⁢(z)∥H˙2⁢δ→H˙2⁢s≤C⁢h2⁢α~

for a constant C is independent of h and z and where

W⁢(z):=|z|-1-α~-s+δ⁢(z⁢Th-I)-1⁢πh⁢(T-Th)⁢(z⁢T-I)-1.

To show this, we write

∥W⁢(z)∥H˙2⁢δ→H˙2⁢s≤|z|-12⁢(1+γ)-s⁢∥(Th-z-1)-1⁢πh∥H˙1-γ→H˙2⁢s⏟=⁣:I⋅∥(T-Th)∥H˙α-1→H˙1-γ⏟=⁣:II

⋅|z|-12⁢(1+α)+δ⁢∥(T-z-1)-1∥H˙2⁢δ→H˙α-1⏟=⁣:III,

where γ:=2⁢α*-α. We estimate the three terms on the right-hand side above separately.

We start with III and use the definition of the dotted spaces (see Section 2.3) to write

∥(T-z-1)-1∥H˙2⁢δ→H˙α-1=supw∈H˙2⁢δ⁡∥T12⁢(1-γ)⁢(T-z-1)-1⁢w∥∥Lδ⁢w∥=supθ∈L2⁡∥T12⁢(1-γ)⁢(T-z-1)-1⁢Tδ⁢θ∥∥θ∥

=∥T1-[12⁢(1+α)-δ]⁢(T-z-1)-1∥.

Applying Lemma A.1 (recall that δ∈[0,12⁢(1+α)] and α∈[0,1] so that 12⁢(1+α)-δ∈[0,1]), we obtain

(A.3)III=|z|-12⁢(1+α)+δ⁢∥(T-z-1)-1∥H˙2⁢δ→H˙α-1≤C,

where C is the constant in (A.1).

To estimate I, we start with the equivalence of norms (3.5) so that

∥(Th-z-1)-1⁢πh∥H˙1-γ→H˙2⁢s≤C⁢∥(Th-z-1)-1∥H˙h1-γ→H˙h2⁢s⁢∥πh∥H˙1-γ→H˙h1-γ.

Whence, the stability of the L2-projection (3.2) together with the equivalence property between dotted spaces and interpolation spaces (Proposition 2.1) as well as the definition of the discrete dotted space norm (3.4) lead to

∥(Th-z-1)-1⁢πh∥H˙1-γ→H˙2⁢s≤C⁢∥Th1-[12⁢(1+γ)+s]⁢(Th-z-1)-1∥.

We recall that α∈(0,1] and γ=2⁢α~-α so that 12⁢(1+γ)+s∈(0,1]. Hence, Lemma A.1 ensures the following estimate:

(A.4)I=|z|-12⁢(1+γ)-s⁢∥(Th-z-1)-1⁢πh∥H˙1-γ→H˙2⁢s≤C.

For the remaining term, Proposition 3.1 with 2⁢s=1-γ gives

II≤C⁢hα+min⁡(α,γ)=C⁢hmin⁡(2⁢α,2⁢α~)=C⁢h2⁢α~.

Combining the above estimate with (A.3) and (A.4) yields (A.2) and completes the proof. ∎

B Sinc Quadrature Lemma

The results of the next lemma are contained in the proof of [3, Theorem 4.1].

Lemma B.1.

Let 0<d<π4 and λ>λ1. Let z⁢(y) be defined by (4.1) and Bd={z∈C:I⁢m(z)<d}. The following assertions hold.

There exists a constant C>0 only depending on λ1, b and d such that
|z⁢(y)-λ|≥C for all ⁢y∈B¯d.
There exists a constant C>0 only depending on λ1, b and d such that
|z′⁢(y)⁢(z⁢(y)-λ)-1|≤C for all ⁢y∈Bd.
There is a constant C>0 only depending on b, d and β such that
ℜ⁢𝔢(z⁢(y)β)≥C⁢2-β⁢eβ⁢|ℜ⁢𝔢y| for all ⁢y∈Bd.

Proof of the Lemma 4.1.

From the expression (4.13) of ℜ⁢𝔢(z⁢(y)) in [3], we deduce that ℜ⁢𝔢(z⁢(y)) is strictly positive for y∈B¯d={w∈ℂ:ℑ⁢𝔪(w)≤d}. It follows from this and part (a) of Lemma B.1 that condition (i) of Definition 4.1 holds for gλ⁢(⋅,t) for λ≥λ1 and t>0.

We now give a proof of (ii) and (iii) of Definition 4.1 simultaneously. Note that part (b) in Lemma B.1 together with (2.5) imply that for y∈B¯d,

|gλ⁢(y,t)|≤C1+tγ⁢|z⁢(y)β|≤C1+tγ⁢|ℜ⁢𝔢(z⁢(y)β)|.

Furthermore, the estimate on ℜ⁢𝔢(z⁢(y)β) in part (c) of Lemma B.1 yields

(B.1)|gλ⁢(y,t)|≤C1+tγ⁢κ⁢2-β⁢eβ⁢|ℜ⁢𝔢y|≤C⁢(β,d,b)⁢t-γ⁢e-β⁢|ℜ⁢𝔢y|.

This guarantees that

∫-dd|gλ⁢(u+i⁢w,t)|⁢𝑑w≤C⁢(β,d,b)⁢t-γ

and

N⁢(Bd)=∫-∞∞(|gλ⁢(u+i⁢d)|+|gλ⁢(u-i⁢d)|)⁢𝑑u≤t-γ⁢C⁢(β,d,b)⁢∫0∞e-β⁢y⁢𝑑y≤C⁢(β,d,b)⁢t-γ

which yield (ii), (iii) and the bound on N⁢(Bd). ∎

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Received: 2017-4-13

Revised: 2017-7-6

Accepted: 2017-8-9

Published Online: 2017-8-26

Published in Print: 2017-10-1

Numerical Approximation of Space-Time Fractional Parabolic Equations

Abstract

A Proof of Lemma 3.2

Lemma A.1.

Proof of Lemma 3.2.

B Sinc Quadrature Lemma

Lemma B.1.

Proof of the Lemma 4.1.

References

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