Error Estimates of Spectral Galerkin Methods for a Linear Fractional Reaction–Diffusion Equation

We consider a fractional diffusion equation with a reaction term in one dimensional space. We first establish the regularity in weighted Sobolev spaces. Then we present an optimal error estimate for a spectral Galerkin method for the equation and a sub-optimal error estimate for a spectral Petrov–Galerkin method. Numerical results suggest that the convergence order in a weighted \(L^2\)-norm is \(2\alpha +1\) for smooth inputs where \(\alpha \) is the order of the fractional Laplacian.

The author would like to thank Professor Hongjie Dong at Brown University for helpful discussion on an early draft of this work. The author also thanks the anonymous reviewers for their valuable comments. This work was partially supported by MURI.

Authors and Affiliations

1. Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA, 01609, USA

   Zhongqiang Zhang

Corresponding author

Correspondence to Zhongqiang Zhang.

Appendix A: Proof of Lemma 2.3

Lemma A.1

[16] Let \(u_p= (1-x^2)^p,\, \left| x\right| \le 1\), \(p>-1\) and \(u_p(x)=0\) when \(\left| x\right| >1\), then for \(x\in (-1,1)\)

Let \(v_p(x)= (1-x^2)^p x,\, \left| x\right| \le 1\) and \(v_p(x)=0\) when \(\left| x\right| >1\), \(p>-1\). Then for \(x\in (-1,1)\)

Here \( B(\cdot , \cdot )\) is the Beta function and the hypergeometric function \({}_2F_1(a,b;c;z) \) is defined for \(|z| < 1\) by the power series

Here \((q)_n\) is the (rising) Pochhammer symbol, which is defined by:

Proof of Lemma 2.3

The conclusion can be shown by induction. By Lemma A.1, for \(n=0,\)\( (-\Delta )^{\alpha /2}[(1-x^2)^{\alpha /2} ] = \Gamma (\alpha +1), \) and for \(n=1\), \( (-\Delta )^{\alpha /2}[(1-x^2)^{\alpha /2} x] = \Gamma (\alpha +2) x.\)

Now suppose that \(k\le n\) the relation holds then when \(k=n+1\), we want

By integration-by-parts formula [22] (or Lemma 2.1), for any \(j\le n\), we have

Here we have used the induction assumption. By the orthogonality of Jacobi polynomials and also by Lemma A.1 that \((-\Delta )^{\alpha /2}[(1-x^2)^{\alpha /2}P_{n+1}^{\alpha /2}(x)]\) is a polynomial of order \(n+1\), then we have

where \(C_{n+1,\alpha }\) is a constant to be determined, depending on \(n+1\) and \(\alpha \). Then we compare the coefficients of leading-order term over both sides and obtain the conclusion at \(k=n+1\).

The constant \(C_{n+1,\alpha }\) can be found as follows. Suppose that \(m=n+1=2k\). We only need to compare the coefficients of the leading order term of both sides of (A.3). By Lemma A.1, it only requires to check the coefficient of leading-order term of \(c_{1,\alpha } B(-\alpha /2, \,p+1) {}_{2}F_{1}(\frac{\alpha +1}{2},-p+\frac{\alpha }{2}; \frac{1}{2}; x^2)\), where \(p=k+\alpha /2\). By the definition of Pochhammer symbol, the coefficient is

Then by the the duplication formula, \( \Gamma (z) \Gamma \left( z + \tfrac{1}{2}\right) = 2^{1-2z} \; \sqrt{\pi } \; \Gamma (2z), \) and the fact that \(\Gamma (1/2)=\sqrt{\pi }\), we have that the coefficient is

Comparing the coefficients of leading order terms of both sides of (A.3), we have \((-1)^{k} C_{m,\alpha } = (-1)^{k+2}\frac{\Gamma (\alpha +1+ m)}{m!}\) and thus \(C_{m,\alpha }= \frac{\Gamma (\alpha +1+ m)}{m!}\). Similarly, we can have the same conclusion when \(m=n+1=2k+1\). \(\square \)

Appendix B: Some Useful Relations of Jacobi Polynomials

The following relations hold for Jacobi polynomials, see e.g. [3, Chapter 2],

The Jacobi polynomial can be represented using Rodrigue’s formula:

The Jacobi polynomial \(P_n^{\alpha ,\beta } (x)\) is a solution of the second order linear homogeneous differential equation

The following is an application of Rodrigue’s formula (B.2) and this equation.

Lemma B.1

For \(\alpha ,\beta >-1\), we have

Lemma B.2

The following relation holds, for \(\beta >-1\)

where \({\widehat{A}}_{n}^{\beta ,\beta } = -\frac{ n+\beta }{2(2n+2\beta +1)}\) and \({\widehat{C}}_{n}^{\beta ,\beta } = \frac{ (n+2\beta +1)(n+2\beta +2)}{2(2n+2\beta +1)(n+\beta +1)}\).

Lemma B.3

For \(\alpha >0\), there is a constant C independent of \(n,\,l,\,\alpha \) such that

Proof

By the Rodrigue’s representation of Jacobi polynomials and integration by parts, we have

Now we compute the integral \((P_{2l}^{\alpha /2+n }, \omega ^{\alpha /2+n-1}).\) From Lemma B.2, we have

Then by induction, we have for \(l\ge 1\)

From Lemma B.2, we have

This leads to

Then we have

Then by (2.5),

Then by the fact which can be proved by Stirling’s formula that

we obtain the conclusion (B.5). \(\square \)

Proof of Lemma 2.5

By Lemma B.2 and Lemma 2.3, we have

Here \({\widehat{A}}_{n}^{\alpha /2-1}\), \({\widehat{C}}_{n}^{\alpha /2-1}\) are from Lemma B.2 and \( D_{n,\alpha }, D_{n-2,\alpha }\) are from Lemma 2.3. \(\square \)

Appendix C: Proof of (3.3)

By (3.2) and Lemma B.2, we have

Here \(E_{k,\alpha }\), \(F_{k,\alpha }\) are from Lemma 2.5 and \({\widehat{A}}_{k}^{\alpha /2-1}\), \({\widehat{C}}_{k}^{\alpha /2-1}\) are defined in Lemma B.2. Thus, multiplying by \(P_{n}^{\alpha /2}\) over both sides of the last equation and by the orthogonality of the Jacobi polynomials, we have

or equivalently for \(n\ge 0\),

Multiplying both sides of (C.1) by \(P_n^{\alpha /2}\), adding all resulting equations, we have

Applying Lemma B.2 leads to

Thus we have from the orthogonality of the Jacobi polynomials that

By the fact that \(P_n^{\beta ,\beta }(-x)= (-1)^nP_n^{\beta ,\beta }(x)\), we have \((P_{k}^{\alpha /2},P_{n}^{\alpha /2-1})_{\omega ^{\alpha /2-1}}=0\) if \(\left| n-k\right| \) is an odd number. Further, the orthogonality of Jacobi polynomials leads to \((P_{k}^{\alpha /2},P_{n}^{\alpha /2-1})_{\omega ^{\alpha /2-1}}=0\) if \(n>k\). Then we obtain (3.3).

Appendix D: Estimates of Products of Functions in Weighted Spaces

Lemma D.1

Let \(u=v\omega ^{\alpha /2}\) with \(v\in B^{\beta }_{ {\alpha /2}}(I) \), \(0<\beta \le \alpha \le 2\). Then \(u\in B^{\beta }_{ {\alpha /2}}(I)\).

Lemma D.2

If \(v\in B^{\beta }_{ {\alpha /2-1}}(I) \), then \(u=v\omega ^{\alpha /2}\in B^{\beta }_{ {\alpha /2-1}}(I)\), when \(0<\beta \le \alpha \le 2\) and \(\beta \ne \alpha /2\).

Lemma D.3

Let \(u=v\omega ^{\alpha /2}\) with \(v\in B^{\beta }_{ {\alpha /2-1}}(I) \), \(0\le \beta \le \alpha +1\), \(0<\alpha \le 2\). Then \(u\in B^{\beta }_{ {\alpha /2}}(I) \) when \(\beta \ne \alpha /2\).

The proofs of these lemmas are similar and we only present the proof of Lemma D.1.

We need an equivalent definition of the weighted Sobolev space. In [17], it is shown that the norm in \(B_{\theta }^s(I)\) (\(s=m+\sigma \), \(0<\sigma <1\) and \(s\ne 1+\theta \) if \(-1<\theta <0\)) is equivalent to the following

where for \(a>1\), \(\Omega _{I,a}\) is a subset of \(I\times I\) and specifically

Here a can be any number larger than 1 and we take \(a=2\).

We also need the following lemma to prove Lemma D.1.

Lemma D.4

Let \(v\in L^{2}_{\omega ^{\gamma +1-\beta +2\theta }}(I)\), where \(\beta <3\), \(\gamma ,\theta \) are real numbers. Then

Here \(g(t,x,\theta )= t^{-2\theta } (1-x)^{\theta }(2t-(1-x))^{\theta }\).

Proof

Note that the integration domain can be split into two parts that \( \Omega _{I,a}\cap \left\{ x>0\right\} \) and \(\Omega _{I,a}\cap \left\{ x<0\right\} .\)

Let consider the case when \(x>0\). Let \(1-x=(1-y)t\). Then \(a^{-1}\le t\le a\), \(\frac{dy}{dt}=\frac{1-x}{t^2}\) and \(\left| x-y\right| = \left| 1-y -(1-x)\right| = \frac{\left| 1-t\right| }{t} (1-x) \). Let \(g(t,x,\theta )= t^{-2\theta } (1-x)^{\theta }(2t-(1-x))^{\theta }=\omega ^{\theta }(y)\). Observe that \(g(1,x,\theta )= \omega ^{\theta }(x)\). The double integral \( \Omega _{I,a}\cap \left\{ x>0\right\} \) becomes

In the last step, we have applied Hardy’s inequality on [1, a] and \([a^{-1},1]\).

Now consider the case when \(x<0\), Let \(1+x=(1+y)t\). Then \(a^{-1}\le t\le a\), \(\frac{dy}{dt}=-\frac{1+x}{t^2}\) and \(\left| x-y\right| = \left| 1+x -(1+y)\right| = \frac{\left| 1-t\right| }{t}(1+x)\). Let \(f(t,x,\theta )= t^{-2\theta } (1+x)^{\theta }(2t-(1+x))^{\theta }=\omega ^{\theta }(y)\). Observe that \(f(1,x,\theta )= \omega ^{\theta }(x)\). The double integral \( \Omega _{I,a}\cap \left\{ x<0\right\} \) becomes

In the last step, we have applied Hardy’s inequality on [1, a] and \([a^{-1},1]\). Note that \(f(t,x,\theta ) = g(t,-x,\theta )\). Observe that \( \int _{a^{-1}}^a (\partial _tf(t,x,\theta ) )^2 \left| 1-t\right| ^{2- \beta } \,dt \le C (1+x)^{2\theta }\). We then obtain the desired conclusion. \(\square \)

Proof of Lemma D.1

If \(\beta =1\) or 2, we can use direct calculation to get the desired results.

When \(0<\beta <1\), \(\int _{I}u^2\omega ^{\alpha /2}\,dx = \int _{I}v^2\omega ^{3\alpha /2}\,dx \le \int _{I}v^2\omega ^{\alpha /2}\,dx\). Moreover,

By Lemma D.4, we have

Since \(\beta \le \alpha \), then \(\int _{I} \omega ^{3\alpha /2- \beta } v^2 \,dx\) is bounded by \(\left\| v\right\| _{\omega ^{\alpha /2}}^2\). Then (D.2) can be bounded by \(C\left\| v\right\| _{\beta ,\omega ^{\alpha /2},B}^2\).

When \(1<\beta \le \alpha <2\). \(\left\| u\right\| _{\omega ^{\alpha /2}}= \left\| \omega ^{\alpha /2}v\right\| _{\omega ^{\alpha /2}}\le \left\| v\right\| _{\omega ^{\alpha /2}}\). As \(\partial _x u = \omega ^{\alpha /2} \partial _x v + (-\alpha x)\omega ^{\alpha /2-1}v \), then

It remains to check the weighted fractional-order norm. Since \(\partial _x u = \omega ^{\alpha /2} \partial _x v + (-\alpha x)\omega ^{\alpha /2-1}v \) and we only need to check that the weighted fractional norms of \(\omega ^{\alpha /2} \partial _x v \) and \( x\omega ^{\alpha /2-1}v \) are bounded. The boundedness of the weighted fractional norm of \(\omega ^{\alpha /2} \partial _x v \) is proved as follows. Let \(\beta =1+\sigma \).

By Lemma D.4, the last term is bounded by \(\int _{I} \omega ^{3\alpha /2+\beta -2\sigma }(\partial _x v)^2\,dx \), which is further bounded by \( \left\| \partial _x v\right\| ^2_{\omega ^{\alpha /2+1} } \) as \(\beta =1+\sigma \le \alpha \).

For \( x\omega ^{\alpha /2-1}v \), we only need to show the regularity of \( \omega ^{\alpha /2-1}v \). Then by the fact that \(\omega ^{\gamma }(y)\le C \omega ^{\gamma }(x) \) on \(\Omega _{I,a}\) for any \(\gamma \), we have

By Lemma D.4, the last term is bounded by \(C \int _{I} \omega ^{3\alpha /2+\beta -2\sigma -2 }(x) v^2(x) \,dx \) and thus it is bounded by \( C \int _{I} \omega ^{\alpha /2 }(x) v^2(x) \,dx. \) Then we have shown that

Then by the definition of \(B^{\beta }_{\alpha /2}(I)\), we have \(u \in B^{\beta }_{\alpha /2}(I)\). \(\square \)

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