    Error estimates for semidiscrete Galerkin and collocation approximations to pseudo-parabolic problems with Dirichlet conditions

This paper is concerned with the numerical approximation of the Dirichlet initial-boundary-value problem of nonlinear pseudo-parabolic equations with spectral methods. Error estimates for the semidiscrete Galerkin and collocation schemes based on Jacobi polynomials are derived.

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1. Introduction

The present paper deals with the numerical analysis to the initial-boundary-value problem (ibvp) of pseudo-parabolic type in

 cvt−(avxt)x=−(αvx)x+βvx+γ,x∈Ω,t>0, (1.1) v(x,0)=v0(x),x∈Ω, (1.2) v(−1,t)=v(1,t)=0,t>0, (1.3)

where are continuously differentiable functions in and bounded above and below by positive constants. The right hand side of (1.1) involves linear and nonlinear terms ; they are assumed to be continuously differentiable functions of and .

In , we perform an extensive study, by computational means, of the use of spectral discretizations, of Galerkin and collocation type, based on Legendre and Chebyshev polynomials for models of type (1.1)-(1.3). The resulting semidiscrete systems are fully discretized there by suitable time integrators, with the aim at overcoming the possible midly stiff character of the ordinary differential problems and the difficulties to simulate nonsmooth data. This computational study is complemented by the present paper with a numerical analysis of the spectral discretization. More specifically, for Galerkin and collocation methods based on a family of Jacobi polynomials (which includes the Legendre and Chebyshev cases described in ) existence of solution of the semidiscrete systems and error estimates in suitable Sobolev norms are derived. As usual for this kind of approaches, the results proved here establish the rate of convergence of the spectral approximation in terms of the regularity of the data of the problem. In particular, they justify those experiments in  concerning spectral convergence in the smooth case.

Pseudo-parabolic equations of the form (1.1), in one or more dimensions, are used for modelling in different areas of Physics and Engineering. Relevant examples are the BBM-Burgers equation, a dissipative modification of the BBM equation for water waves, , and the pseudo-parabolic Buckley-Leverett equation describing two-phase flow in porous media, . We refer to the rich bibliography on it commented in .

The mathematical theory of pseudo-parabolic equations can be covered by [40, 41, 52, 53, 27, 13, 17, 20, 50]. Existence and uniqueness of weak solutions to nonlinear pseudo-parabolic equations are proved in , whereas the existence of weak solutions for degenerate cases is studied in [44, 43]. A homogenization of a closely related pseudo-parabolic system is considered in . Traveling wave solutions and their relation to non-standard shock solutions to hyperbolic conservation laws are investigated in [21, 23] for linear higher order terms. Uniqueness of weak solutions for a pseudo-parabolic equations modelling flow in porous media can be found in [18, 36, 17]. In , the authors study existence and uniqueness of weak solutions of the initial and boundary value problem for a fourth-order pseudo-parabolic equation with variable exponents of non-linearity, along with a long-time behaviour of weak solutions. Finally, existence of weak solutions for a nonlocal pseudo-parabolic model for Brinkman two-phase flow model in porous media has been recently established, .

Concerning the numerical approximation of equations of the form (1.1), the literature contains many references involving finite differences, [4, 5, 54, 6, 22, 28], as well as finite elements and finite volumes, [38, 45, 2, 56]

. We also mention some convergence results. First, stability and convergence of difference approximations to pseudo-parabolic partial differential equations is discussed in

[29, 30] and the time stepping Crank-Nicolson Galerkin method to approximate several nonlinear Sobolev-type problems is analyzed in [26, 25]. Of particular relevance for the present study is the finite element approach for the nonlinear periodic-initial-boundary-value problem, , where Arnold and collaborators obtain optimal error estimates, in and norm, of a standard Galerkin method with continuous piecewise polynomials, and a nodal superconvergence. Moreover, Fourier spectral methods of Galerkin and collocation type for quasilinear pseudo-parabolic equations are analysed in . This is, to our knowledge, the main reference about the use of spectral methods to approximate Sobolev equations. More recent convergence results can be found in , where an analysis of a linearization scheme for an interior penalty discontinuous Galerkin for a pseudo-parabolic model in porous media applications is considered. High-order finite differences are employed in 

and B-spline quasi-interpolation methods in

. In addition, an adaptive mesh approach for pseudo-parabolic-type problems is introduced in  and a Meshless RBFs method is considered in 

. Finally, unconditionally stable vector splitting schemes for pseudo-parabolic equations are constructed and analyzed in

. It is worthwhile to mention that standard operator splitting may fail to capture the correct behavior of the solutions for pseudo-parabolic type differential models. In , the authors presented a non-splitting numerical method which is based on a fully coupled space-time mixed hybrid finite element/volume discretization approach to account for the delicate nonlinear balance between the hyperbolic flux and the pseudo-parabolic term linked to the full pseudo-parabolic differential model.

The structure of the paper is as follows. Section 2 is devoted to some theoretical aspects of (1.1)-(1.3) as the weak formulation (already mentioned in  for Legendre and Chebyshev cases) and assumptions on well-posedness. These preliminaries also include a summary on inverse inequalities, as well as projection and interpolation error estimates for the family of Jacobi polynomials under consideration. The contents of Section 2 will be used to the numerical analysis of the spectral Galerkin approximation in Section 3, and the collocation approximation in Section 4. Both contain, under suitable hypotheses on the data of the problem, results on the existence of numerical solution and convergence to the solution of (1.1)-(1.3). Concluding remarks and perspectives for future work are outlined in Section 5.

We now describe the main notation used throughout the paper. For positive integer , denotes the normed space of -functions on with as associated norm, while for nonnegative integer , is the space of -th order continuously differentiable functions on .

Let and define the Jacobi weight function

 w(x)=wμ(x)=(1−x2)μ,x∈Ω. (1.4)

( corresponds to the Legendre case and to the Chebyshev case.) Then will denote the space of squared integrable functions with respect to the weighted inner product

 (ϕ,ψ)w=∫1−1ϕ(x)ψ(x)w(x)dx,ϕ,ψ∈L2w, (1.5)

and associated norm . For the Sobolev spaces integer (where ) the corresponding norm will be denoted by

 ||ϕ||2k,w=k∑j=0||djdxjϕ||20,j.

We will also consider the spaces of functions such that . For , (and ) are defined by interpolation theory, . Note that in the case of the Legendre approximation () the spaces are the standard Sobolev spaces .

For an integer , will stand for the space of polynomials of degree at most on and

 P0N={p∈PN/p(−1)=p(1)=0}.

If and , stands for the space of functions on with norm . For an integer , the space of -th order continuously differentiable functions , where or , will be denoted by . Additionally, if , will stand for the normed space of functions with associated norm

We also denote by the space of functions with finite norm

 ||u||L∞(0,T,X)=esssupt∈(0,T)||u(t)||s,w,

where stands for the essential spectrum. Furthermore, (resp. ) will stand for the space of continuously differentiable (resp. uniformly bounded, continuously differentiable) functions in .

The analysis of the collocation methods requires the introduction of discrete norms. Let be the nodes and weights of the Gauss-Lobatto quadrature related to , [42, 16, 12]. For continuous on , the discrete inner product based on the Gauss-Lobatto data is denoted by

 (ϕ,ψ)N,w=N∑j=0ϕ(xj)ψ(xj)wj, (1.6)

with associated norm . We recall that, 

 (ϕ,ψ)N,w=(ϕ,ψ)w,

if . The equivalence of the norms and when , established in the following lemma, was proved in  for the case of Legendre and Chebyshev weights and in  for (1.4) with .

Lemma 1.1.

Let be an integer. Then there exist positive constants , independent of , such that for any

 C1||ϕ||0,w≤||ϕ||N,w≤C2||ϕ||0,w.

Finally will be used to denote a generic, positive constant, independent of and , but that may depend on (this will be denoted by ).

2. Preliminaries

2.1. Weak formulation

The analysis of the spectral discretizations that will be made below requires some hypotheses, properties and technical results concerning (1.1)-(1.3) and the approximation in weighted norms. From now on we will fix and consider the weight (1.4). The first property to be mentioned is the weak formulation of (1.1)-(1.3), cf. 

 A(vt,ψ)=B(v,ψ),ψ∈H1w,0 (2.1)

with and

 A(ϕ,ψ) = (cϕ,ψ)w+La(ϕ,ψ), (2.2) B(ϕ,ψ) = Lα(ϕ,ψ)+(β(ϕ)ϕx,ψ)w+(γ(ϕ),ψ)w,ϕ,ψ∈H1w,0,

where, for

 Ld(ϕ,ψ)=∫1−1dϕx(ψw)xdx.

Since is bounded above and below by positive constants, then is equivalent to

 L(ϕ,ψ)=∫1−1ϕx(ψw)xdx, (2.3)

and therefore, [11, 16, 12], the bilinear form in (2.2) is continuous in and elliptic in , that is, there are positive constants such that for all

 |A(ϕ,ψ)| ≤ C1(||ϕ||0,w||ψ||0,w+||ϕx||0,w||ψx||0,w) ≤ C1||ϕ||1,w||ψ||1,w,ϕ∈H1w,ψ∈H1w,0, A(ϕ,ϕ) ≥ C2||ϕ||21,w,ϕ,ψ∈H1w,0.

The weak formulation (2.1) is used to assume well-posedness of (1.1)-(1.3), according to the following results, cf. [52, 8]

Theorem 2.1.

Let and assume that , . Given , then there is a unique solution of (2.1) with bounded by a constant depending only on and the data of the problem. Furthermore, if with integer, , then for all and

 ||v|||L∞(0,T,Hkw)+||vt|||L∞(0,T,Hkw)≤C, (2.4)

where is a constant depending only on and the data of the problem.

2.2. Projection and interpolation errors with Jacobi polynomials

Here we collect several results concerning projection and interpolation errors with respect to the weighted inner product (1.5) and that will be used below. We refer to, e. g., [32, 42, 39, 11, 16, 15, 12, 51] for details and additional properties.

The estimates in the weighted Sobolev spaces , concern the use of the family of Jacobi polynomials , which are orthogonal to each other in . Particular cases such as Legendre and Chebyshev families correspond to and , respectively. Most properties of this Jacobi family (a particular one of the more general Jacobi polynomials , orthogonal in with ) are extension of the corresponding properties of the Legendre family, [11, 12].

We start with projection errors. Let be an integer, and let be the orthogonal projection of with respect to the inner product (1.5), and be the orthogonal projection of with respect to the inner product in

 [ϕ,ψ]w=∫1−1ϕ′(x)ψ′(x)w(x)dx.

Then we have, [11, 14, 12]

 ||v−PNv||0,w ≤ CN−s||v||s,w,v∈Hsw,s≥0, (2.5) ||v−PNv||r,w ≤ CN2r−1/2−s||v||s,w,v∈Hsw,1≤r≤s,

and for

 ||v−P10Nv||1,w+N||v−P10Nv||0,w ≤ CN1−s||v||s,w, (2.6) s≥1,−1<μ≤0, ||v−P10Nv||1,w+N1−μ||v−P10Nv||0,w ≤ CN1−s||v||s,w, (2.7) s≥1,0<μ≤1.

In the Legendre and Chebyshev cases, sharper estimates hold, see [15, 16].

A third projection operator used below concerns the bilinear form given by (2.2). If then the orthogonal projection of with respect to is defined as such that

 A(¯¯¯v−v,ψ)=0,ψ∈P0N. (2.8)

For this projection, we have, 

 ||v−¯¯¯v||1,w+N||v−¯¯¯v||0,w≤CN1−m||v||m,w, (2.9)

for . Furthermore, a generalized estimate can be obtained as follows. If , let be a polynomial such that, 

 ||uN−v||k,w≤CNk−m||v||m,w.0≤k≤2. (2.10)

By using (2.9), (2.10) and the inverse inequalities, 

 ||ψ||s,w≤CN2(s−r)||ψ||r,w,ψ∈PN,0≤r≤s,

we have

 ||v−¯¯¯v||2,w ≤ ||uN−v||2,w+||uN−¯¯¯v||2,w (2.11) ≤ CN2−m||v||m,w+CN2||uN−¯¯¯v||1,w ≤ CN2−m||v||m,w+CN2(||uN−v||1,w+||v−¯¯¯v||1,w) ≤ CN3−m||v||m,w.

Let be an integer, and let denote the interpolant polynomial of on based on the Gauss-Lobatto-Jacobi nodes. The following estimates for the interpolation errors can be seen in [11, 12]: for

 ||v−INv||r,w ≤ CNr−s||v||s,wv∈Hsw, (2.12) 0≤r≤1,s>sup{1+r2,1+r+μ2}.

Finally, an additional estimate comparing the continuous and discrete inner products will be necessary: if and , then

 |(f,ϕ)w−(f,ϕ)N,w| ≤ C(||f−PN−1f||0,w (2.13) +||f−INf||0,w)||ϕ||1,w,

where is given by (1.6), .

3. Spectral Galerkin approximation

Let be an integer, . The semidiscrete Galekin approximation is defined as the function satisfying, cf. 

 A(vNt,ψ) = B(vN,ψ),ψ∈P0N, (3.1) A(vN(0),ψ) = A(v0,ψ),ψ∈P0N. (3.2)
Remark 3.1.

In what follows, we will make use of the two identities below (see, e.g., [8, 48]). Let a function of . Then

 F(v)−F(z) = (v−z)F∗(v,z), (3.3) F(v)vx−F(z)zx = F(v)(v−z)x+zx(v−z)F∗(v,z), (3.4) F∗(v,z) = ∫10Fu(v+τ(z−v))dτ.

Local existence and uniqueness of (3.1), (3.2

) are ensured by standard theory of ordinary differential equations (ode) when (

3.1) is considered as a finite system for the coefficients of in some basis of , by using the property of ellipticity of and continuity of . Concerning this last point, some estimates on will be required in order to prove a global existence result. This is discussed in the following remark.

Remark 3.2.

We write in (2.2) in the form

 B(ϕ,ψ) = B1(ϕ,ψ)+B2(ϕ,ψ)+B3(ϕ,ψ) B1(ϕ,ψ) = ∫1−1α(ϕ)ϕx(ψw)xdx, B2(ϕ,ψ) = ∫1−1β(ϕ)ϕx(ψw)dx, B3(ϕ,ψ) = ∫1−1γ(ϕ)(ψw)dx,

and assume that . Then, using continuity of (2.3), there are constants such that

 |B1(ϕ,ψ)| ≤ α1||ϕ||1,w||ψ||1,w, |B2(ϕ,ψ)| ≤ β1||ϕ||1,w||ψ||1,w.

As far as is concerned, from (3.3), (3.4) we can find a function , bounded on , and a constant such that

 |B3(ϕ,ψ)|≤(δ+γ1||ϕ||1,w)||ψ||1,w.

Global existence and uniqueness for (3.1), (3.2) and the convergence to the solution of (2.1), (2.2) are proved in the following result.

Theorem 3.1.

For all , there is a unique solution of (3.1), (3.2) satisfying

 ||vN||L∞(0,T,H1w)≤C, (3.5)

for some constant depending on . Furthermore, assume that , , with . Then

 ||vN−v||L∞(0,T,L2w) ≤ CN−m, (3.6) ||vN−v||L∞(0,T,H1w) ≤ CN1−m, (3.7)

for some constant which depends on but not on .

Proof.

Following previous approaches, [8, 48], we first assume that . By using property of ellipticity of and Remark 3.2, we set in (3.1) and have

 ||vNt||1,w≤C||vN||1,w+δ(t),

for some constant . Then

 ||vN||1,w = ||vN(0)+∫t0vNt(s)ds||1,w (3.8) ≤ ||vN(0)||1,w+C∫t0||vN(s)||1,wds+||δ||L1(0,T).

From (3.2) with and properties of continuity and coercivity of we have . This and Gronwall’s lemma applied to (3.8) imply the existence of for all and (3.5).

As far as the error estimates are concerned, let be the projection defined in (2.8) and

 η=¯¯¯v−v,eN=vN−v,ξN=¯¯¯v−vN=η−eN∈P0N. (3.9)

Note that, due to (2.8), holds. Thus, (1.1) and (3.1) imply, for

 A(ξNt,ψ)=−A(eNt,ψ)=−(B(vN,ψ)−B(v,ψ)). (3.10)

The right hand side of (3.10) is written as

 B(vN,ψ)−B(v,ψ) = ˜B1+˜B2+˜B3, (3.11) ˜B1 = ∫1−1(α(v+eN)(v+eN)x−α(v)vx)(ψw)xdx, ˜B2 = ∫1−1(β(v+eN)(v+eN)x−β(v)vx)(ψw)dx, ˜B3 = ∫1−1(γ(v+eN)−γ(v))(ψw)dx.

We use (3.3), (3.4), the hypothesis and Theorem 2.1 to have

 |˜B1| = |∫1−1(α(vN)eNx+vxeNα∗(vN,v))(ψw)xdx| ≤ C||eN||1,w||ψ||1,w, |˜B2| = |∫1−1(β(vN)eNx+vxeNβ∗(vN,v))(ψw)dx| ≤ C||eN||1,w||ψ||1,w, |˜B3| = |∫1−1eNγ∗(vN,v))(ψw)dx|≤C||eN||0,w||ψ||0,w.

Therefore

 |B(vN,ψ)−B(v,ψ)|≤C||eN||1,w||ψ||1,w, (3.12)

for and with depending on . Then, taking and using the coercivity of , (3.10) and (3.12) we obtain

 ||ξNt||1,w≤C||eN||1,w. (3.13)

Since (3.2) implies that and therefore , then writing yields

 ||ξN(t)||1,w = ||∫t0ξNt(s)ds||1,w ≤ C∫t0(||ξN(s)||1,w+||η(s)||1,w)ds.

Therefore, (3.7) holds from Gronwall’s lemma, the property and Theorem 2.1.

We now prove the estimate (3.6). Lax-Milgram theorem, , ensures the existence of such that, [11, 39]

 A(ψ,φ)=(ξNt,ψ)0,w,ψ∈H1w,0. (3.14)

Actually ( see ) and

 ||φ||2,w≤C||ξNt||0,w. (3.15)

We take in (3.14) and use (3.10) to estimate, (cf. )

 ||ξNt||20,w ≤ A(ξNt,φ)=A(ξNt,φ−P10Nφ)+A(ξNt,P10Nφ) (3.16) = A(ξNt,φ−P10Nφ)−B(eN,P10Nφ) = A(ξNt,φ−P10Nφ)+B(eN,φ−P10Nφ) −B(eN,φ).

Now, continuity of , (2.6), (2.7) and (3.15) imply

 A(ξNt,φ−P10Nφ) ≤ C||ξNt||1,w||φ−P10Nφ||1,w (3.17) ≤ CN−1||ξNt||1,w||φ||2,w ≤ CN−1||ξNt||1,w||ξNt||0,w.

On the other hand, Remark 3.2, (2.6), (2.7) and (3.15) lead to

 |B(eN,φ−P0Nφ)| ≤ C||eN||1,w||φ−P0Nφ||1,w (3.18) ≤ CN−1||eN||1,w||φ||2,w ≤ CN−1(||eN||1,w+ ||δ−P10Nδ||0,w)||ξNt||0,w.

We now consider defined in (3.11). Integrating by parts, we can write, [8, 48]

 ˜B1(eN,φ) = ∫1−1eN((−αx−αu(vN)vNx+vxα∗)(φw)x −α(vN)(φw)xx)dx,

and several applications of Hardy inequality, see e. g.  (this is not necessary of course in the Legendre case ), hypothesis , (3.5) and Theorem 2.1 imply

 |˜B1(eN,φ)| ≤ C||eN||0,w||φ||2,w. (3.19)

Similarly, we write

 ˜B2(eN,φ)=∫1−1eN((−βx−βu(vN)vNx+vxβ∗)(φw))dx,

and hypothesis , (3.5), Theorem 2.1 and continuity of in (2.3) lead to

 |˜B2(eN,φ)| ≤ C||eN|