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Variational principles for advection–diffusion problems
Computers and Mathematics with Applications (
)
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Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Variational principles for advection–diffusion problems Giles Auchmuty Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA
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Article history: Available online xxxx Keywords: Advection–diffusion equations Mixed boundary conditions Variational principles
a b s t r a c t Variational principles for linear and semilinear advection–diffusion problems with velocity field given by potential flow are described and analyzed. Mixed Dirichlet and prescribed flux conditions are treated. Existence and uniqueness results are proved and equivalent integral operator equations are found. A positive multiplier function related to the potential of the flow is used to change the system to divergence form. The dependence of the solution on inhomogeneous flux boundary data is determined. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction This paper describes variational principles and some results about the solutions of linear and semilinear advection– diffusion equations in bounded regions and subject to mixed Dirichlet and flux boundary conditions. Google searches provide a tremendous number of papers on the numerical simulation of these problems; a paper by Brooks and Hughes [1] has over 5000 citations in Google scholar. These equations arise in some very disparate areas of science and engineering. Most of the papers describe the numerical computation of solutions under different assumptions on the data. There has also been extensive study of algorithms and procedures for efficiently simulating the observed phenomenology. See chapter 5 of Glowinski [2] for a recent discussion of least squares and other algorithms for these systems. The numerical simulations indicate that, when the advection term is significant, it is better to use methods that respect properties of the velocity field. General existence results for these systems are known based on standard elliptic theory. See section 6.9 of Attouch, Buttazzo and Michaille [3] where existence–uniqueness is proved under an assumption of ‘‘slow flow’’ and zero Dirichlet boundary conditions. Some texts on finite element simulations also prove some existence results. However there has been relatively little mathematical analysis of problems with physically interesting non-zero boundary conditions that help understand the results observed when the velocity terms are significant. Here the construction and properties of variational principles for the solutions will be treated for velocity fields that are gradient flows. This boundary value problem is to find solutions of
− ∆u + a · ∇ u = f (x, u) on Ω
(1.1)
subject to mixed Dirichlet and flux boundary conditions u=0
on
˜ and Dν u = g(x) Σ
on Σ .
(1.2)
Here a : Ω → Rn is assumed to be a gradient field on the bounded region Ω ⊂ Rn with Lipschitz boundary ∂ Ω . Σ is a ˜ = ∂ Ω \ Σ , ν is the unit outward normal at a point on the boundary and f , g are proper open subset of the boundary, Σ prescribed functions on Ω × R, Σ respectively. Further conditions on these functions will be specified below. E-mail address: [email protected]. https://doi.org/10.1016/j.camwa.2017.09.023 0898-1221/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: G. Auchmuty, Variational principles for advection–diffusion problems, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.09.023.
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(1.1)–(1.2) are of the form treated by Ortiz [4] who described some quite different variational principles for this problem. Note that if a is a non-zero vector field, then (1.1) is not in divergence form, nor is it a potential equation, that holds at the critical points of a Gateaux-differentiable function on a Sobolev space. Nevertheless, under the condition that a = − ∇ ϕ is a gradient field, there is a functional E on a natural Hilbert Sobolev 1 (Ω ) whose critical points are precisely the solutions of this system. This functional involves a multiplier (or space HΣ integrating factor) determined by the potential of the flow field a. It appears that this multiplier plays a role analogous to that of preconditioners for the numerical simulation of solutions. The variational principle is used to prove an existence–uniqueness result for the system subject to simple conditions on the source (or reaction) term and standard assumptions on the region and the boundary conditions. The linear case is described carefully in Section 3 and an equivalent integral formulation is outlined. Section 4 provides representations and approximations of solutions that satisfy the inhomogeneous boundary data. These use a class of mixed Steklov eigenfunctions for the problem. 2. Assumptions and spaces For the most part, the notation is standard and similar to that used in [3]. The assumptions on the region Ω , its boundary ˜ are appropriate for the use of finite element models. ∂ Ω and the subsets Σ , Σ (B1) Ω is a bounded connected open set in Rn whose boundary ∂ Ω is the union of a finite number of disjoint closed Lipschitz surfaces; each surface having finite surface area and a unit outward normal ν (.) defined σ a.e.. (When n = 2, these are Lipschitz curves of finite length). ˜ have strictly positive surface area, σ (∂ Σ ) = 0 and g ∈ LqT (Σ , dσ ) with (B2) Σ is a nonempty open subset of ∂ Ω , Σ and Σ qT = 2(1 − 1/n) for n ≥ 3 (qT > 1 when n = 2). The advection term is taken to be a gradient field a(x) := − ∇ ϕ (x) with potential ϕ . A necessary condition for this is that curl a ≡ 0 on Ω when n = 2 or 3. The potential ϕ should satisfy (B3) ϕ is a Lipschitz (W 1,∞ ) function with ϕ (x) ≥ 0 on Ω . Observe that ideal fluid flows are potential flows and the positivity requirement on ϕ holds without loss of generality since adding a constant to a potential function does not change the gradient. When a(x) ≡ a is a constant ∑n field2 the potential has the form ϕ (x) = a · x + a0 . When a(x) = 2 c · x + c0 is linear, the potential is quadratic; ϕ (x) = j=1 cj xj + c0 · x + a0 with a0 chosen so ensure the positivity of ϕ on the compact set Ω . 1 Let Γ : H 1 (Ω ) → L2 (∂ Ω ) be the boundary trace operator and HΣ (Ω ) be the subspace of H 1 (Ω ) of all functions that ˜ . This is a closed subspace of H 1 (Ω ) and will be a real Hilbert space with respect to the inner satisfy Γ u(x) = 0 σ a.e. on Σ product
⟨u, v⟩ :=
∫
∇ u · ∇ v dx
Ω
since Σ is non-empty.
1 A weak formulation of the problem is to find u ∈ HΣ (Ω ) that satisfies
a(u, v ) :=
∫ Ω
[ ∇ u · ∇ v + (a · ∇ u ) v ] dx = m(u, v ) for all v ∈ HΣ1 (Ω ).
(2.1)
Here dx denotes Lebesgue measure and m(u, v ) :=
∫ Ω
f (x, u) v dx +
∫ Σ
g(x) v dσ
(2.2)
with dσ being surface area measure on the boundary ∂ Ω . Define χ (x) := eϕ (x) then χ ∈ W 1,∞ (Ω ) and χ (x) ≥ 1 on Ω ; χ will be called a multiplier for this variational problem. It is easily verified that u solves (1.1) provided
− div( χ ∇ u ) = χ (x) f (x, u) on Ω .
(2.3)
This equation is in divergence form so there is a variational principle for its solutions. Consider the problem (P ) of minimizing 1 1 E on HΣ (Ω ) where E : HΣ (Ω ) → R is defined by
∫
χ | ∇ u | − 2 F (x, u) dx − 2
[
E (u) := Ω
2
]
∫ Σ
χ g u dσ
(2.4)
∫s
with F (x, s) := 0 f (x, t) dt. Note that this functional involves the advection field solely through the multiplier χ (x). To obtain existence results for this variational problem we require the following condition on the source term f and its antiderivative F. (B4) Assume f (., s) is Borel measurable on Ω for each s ∈ R, f (x, .) is continuous on R for almost all x ∈ Ω and for each ϵ > 0 there is a C (ϵ ) such that F (x, s) ≤ C (ϵ ) + ϵ s2 on Ω × R. This is the condition that f is a Caratheodory function whose indefinite integral is sub-quadratic in s. In particular it holds when f is bounded and continuous on Ω × R. Please cite this article in press as: G. Auchmuty, Variational principles for advection–diffusion problems, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.09.023.
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1 Theorem 2.1. Assume (B1)–(B4) hold, then E is weakly lower semi-continuous (w.l.s.c) and coercive on HΣ (Ω ), and there are 1 minimizers of E on H (Ω ). The minimizers satisfy
∫ Ω
χ [ ∇ u · ∇ v − f (x, u) v ] dx −
∫ Σ
χ g v dσ = 0 for all v ∈ HΣ1 (Ω ).
(2.5)
1 (Ω ) so they are w.l.s.c. To prove Proof. The first and last terms in the expression for E are convex and continuous on HΣ 1 that second term is w.l.s.c. let uk be a sequence of functions that converges weakly to a limit u˜ in HΣ (Ω ). Then uk converges 2 1 strongly to u˜ in L (Ω ) from Rellich’s theorem and thus in L (Ω ) as Ω is bounded. Decompose F (x, s) = F+ (x, s) − F− (x, s) into its ∫ positive and negative parts ∫with F+ (x, s) := max(F (x, s), 0). From theorem 2.3 of Ambrosetti and Prodi, [5], it follows that Ω F+ (x, uk ) dx converges∫ to Ω F+ (x, u˜ ) dx as k → ∞ ∫ in view of the upper bound requirement of (B4). Theorem 2.3 of Giaquinta [6] implies that Ω F− (x, u˜ ) ≤ lim infk→∞ Ω F− (x, uk ) dx so this functional is w.l.s.c. Thus each term in the expression for E is w.l.s.c. ∫ ∫ ˜ has positive surface measure there is a c0 > 0 such that Since Ω is bounded and Σ | ∇ u |2 dx ≥ c0 Ω u2 dx for all Ω 1 1 (Ω ) and there will be a minimizer of E as claimed. (Ω ). Choose ϵ < c0 in (B4), then the functional E is coercive on HΣ u ∈ HΣ 1 1 When v ∈ HΣ (Ω ) ∩ C (Ω ), the first variation of E at u ∈ HΣ (Ω ) is δ E (u, v ) = limt →0 t −1 [ E (u + t v ) − E (u) ] . This exists 1 and is given by the left hand side of (2.5). At the minimizer it will be zero, so (2.5) holds by density for all v ∈ HΣ (Ω ). □
Observe that (2.5) is a weak version of Eq. (2.3) so the minimizers of E on H 1 (Ω ) are weak solutions of (2.3)–(1.2). The next section investigates the case where f (x, s) is an affine function in more detail. 3. Solution operators for linear advection–diffusion Here some properties of the solutions of this problem in the case where the original system (1.1) is linear will be described. Consider the case where the equation is
− ∆u − ∇ ϕ · ∇ u + c u = f0 (x)
on Ω .
(3.1)
The analog of Eq. (2.3) now is
− div( χ ∇ u ) + χ c u = χ (x) f0 (x)
on
Ω.
(3.2)
Consider the problem of minimizing E (.) on HΣ (Ω ) with E (u) = E1 (u) + F (u) where 1
∫ E1 (u) :=
χ | ∇ u | dx, 2
Ω
∫
χ c u − 2 f0 u dx − 2
[
F (u) := Ω
2
]
∫ Σ
χ g u dσ .
(3.3)
Note that this differs from the energy functions for standard Poisson type problems by the inclusion of the weight function χ := eϕ (x) ≥ 1 in each integral. To prove existence and uniqueness the following assumptions on f0 , c are imposed. (B4L) Assume f0 , c are Borel measurable functions with c ≥ 0 in Ln/2 (Ω ), f0 ∈ LqS (Ω ), qS = 2n/(n + 2) when n ≥ 3, or else qS > 1 when n = 2. This variational problem now has the following properties — so its minimizer is a weak solution of (3.2). Theorem 3.1. Assume (B1)–(B3) and (B4L) hold, then F is weakly lower semicontinuous and the functional E is strictly convex, 1 1 continuous and coercive on HΣ (Ω ). The unique minimizer of E is the unique solution in HΣ (Ω ) of
∫ Ω
χ [ ∇ u · ∇ v + ( c u − f0 )v ] dx −
∫ Σ
χ g v dσ = 0 for all v ∈ HΣ1 (Ω ).
(3.4)
Proof. When (B4L) holds, then the Sobolev imbedding theorem implies that each term in F is well-defined and continuous 1 on HΣ (Ω ). The quadratic term is convex as c ≥ 0 so the functional E is coercive and strictly convex. Hence the functional attains a unique minimizer. The proof that this minimizer satisfies (3.2) is similar to that of Theorem 2.1. □ 1 Observe that (3.4) is a weak formulation of (3.2). When g ≡ 0 let G (f0 ) be the minimizer of this E on HΣ (Ω ) and the minimizer when f ≡ 0 be denoted Gb (g). Consider the restriction of these solution operators to data f0 , g in L2 (Ω ), L2 (∂ Ω ) respectively. 1 Theorem 3.2. Assume that (B1)–(B3) and (B4L) hold, then the maps G , Gb are continuous linear maps into HΣ (Ω ). They are compact linear transformations of L2 (Ω ), L2 (∂ Ω ) respectively into L2 (Ω ). 1 Proof. The verification that G , Gb are linear transformations is standard. They are bounded maps from LqS (Ω ) to HΣ (Ω ) 1 qT and from L (Σ ) to HΣ (Ω ) with the choice of qS , qT from conditions (B2) and (B4L) upon using the Sobolev imbedding and 1 trace theorems and coercivity inequalities. Hence they are continuous linear maps of G , Gb respectively into HΣ (Ω ). From 1 2 Rellich’s theorem, the imbedding of HΣ (Ω ) into L (Ω ) is compact, so by composition both G , Gb will be compact maps from L2 (Ω ), L2 (∂ Ω ) since qS < 2 and qT < 2 for all n ≥ 2. □
Please cite this article in press as: G. Auchmuty, Variational principles for advection–diffusion problems, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.09.023.
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Since these are compact linear transformations, they will have good finite rank approximations and may be written as
∫ (G f0 )(x) :=
G(x, y) f0 (y) dy
Ω
∫ and
(Gb g)(x) :=
Σ
Gb (x, z) g(z) dσ (z)
(3.5)
where the integral kernels G, Gb are limits of appropriate Galerkin approximations. In particular the kernel G can be approximated using eigenfunctions and eigenvalues of the operator Lu := χ c u − div(χ ∇ u ) subject to zero-Dirichlet ˜ and zero-Neumann conditions on Σ . boundary conditions on Σ The solutions of the system (3.3) can now be shown to be the solutions u ∈ L2 (Ω ) of the linear compact operator equation
∫ u(x) = Σ
Gb (x, z) g(z) dσ (z) +
∫ Ω
G(x, y) f0 (y) dy.
(3.6)
This equation is of a form for which there is an extensive analytic and computational theory. In particular it may be used to study the dependence of the solutions on the interior sources f and/or the boundary data g through the subset Σ of the boundary ∂ Ω . 4. Dependence of solutions on the boundary flux Physically, an important question for these problems is the dependence of the solutions on the boundary flux g. From (B2) and Theorem 2.1, there will be an H 1 -solution of these problems whenever g ∈ L4/3 (Σ ) for problems posed on regions in R3 . In this section, an explicit representation of a solution of the homogeneous equation div(χ ∇ u ) = 0 on Ω subject to the boundary conditions (1.2) will be described. The expressions are obtained via similar methods to those described in [7] for some different mixed boundary value problems. The boundary conditions associated with the minimizers of E defined by (2.4) are weak forms of (1.2). That is we seek 1 properties of solutions u˜ ∈ HΣ (Ω ) of the system
∫ Ω
χ ∇ u · ∇ v dx =
∫ Σ
g v χ dσ
for all
v ∈ HΣ1 (Ω ).
(4.1)
Some different representations and formulae for this solution may be found depending on how terms involving the function c are treated. A choice that involves the velocity potential ϕ , but is independent of the source term, may be found by using mixed 1 Steklov (mS-)eigenfunctions that produce orthogonal bases of HΣ (Ω ) with respect to the inner product
⟨u, v⟩χ :=
∫
χ ∇ u · ∇ v dx +
Ω
∫ Σ
u v χ dσ .
(4.2)
1 1 This is called the χ -inner product on HΣ (Ω ) and it is straightforward to verify that HΣ (Ω ) is a real Hilbert space 1 with respect to this inner product. A function s ∈ HΣ (Ω ) is a mS-eigenfunction corresponding to an eigenvalue λ provided it is a non-zero solution of the system
a(u, v ) :=
∫ Ω
χ ∇ s · ∇ v dx = λ
∫ Σ
χ s v dσ
for all
v ∈ HΣ1 (Ω ).
(4.3)
Such functions s are weak solutions of the system div( χ∇ s ) = 0
on Ω
with s = 0
on
˜, Σ
Dν s = λ s
on
Σ.
(4.4)
1 Let V0 to be the subspace of HΣ (Ω ) of functions u that satisfy
a(u, v ) =
∫ Ω
χ ∇ u · ∇ v dx = 0 for all v ∈ H01 (Ω ).
(4.5)
These functions are weak solutions of the differential equation in (4.4) so V0 may be regarded as the subspace of χ -harmonic 1 functions in HΣ (Ω ). Then 1 HΣ (Ω ) = H01 (Ω )⊕χ V0
(4.6)
with ⊕χ denoting orthogonal sum with respect to the χ -inner product (4.2). This has a similar form to those studied first in [8]. The mS-eigenproblem is of the type studied in Auchmuty ∫ [9] and the eigenfunctions satisfying (4.3) are in V0 . In the 1 notation of [9], a is as above while V = HΣ (Ω ), m(u, v ) := Σ χ u v dσ . Theorem 3.1 then yields that there is a least, strictly 1 positive, mS-eigenvalue λ1 of (4.3) and a corresponding mS-eigenvalue s1 ∈ HΣ (Ω ). Moreover the coercivity inequality
∫
χ | ∇ u | dx ≥ λ1 2
Ω
∫ Σ
χ u2 d σ
(4.7)
Please cite this article in press as: G. Auchmuty, Variational principles for advection–diffusion problems, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.09.023.
G. Auchmuty / Computers and Mathematics with Applications (
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1 holds for all u ∈ HΣ (Ω ). Let L2 (Σ , χ dσ ) be the usual Lebesgue space with the inner product
⟨u, v⟩χ ,Σ :=
∫ Σ
u v χ dσ .
(4.8)
Results from section 4 of [9] show that there is an increasing sequence Λ := {λj : j ≥ 1} of mS-eigenvalues and associated mS-eigenfunctions S := {sj : j ≥ 1} with (R1) λj → ∞ as j → ∞, (R2) the boundary traces of the functions in S have support in Σ and are an orthonormal basis of L2 (Σ , χ dσ ). (R3) S is a maximal χ -orthogonal set in V0 . In view of (R2), elementary Hilbert space theory says that g ∈ L2 (Σ , χ dσ ) has a Fourier expansion of the form g(z) =
∞ ∑
gˆj sj (z)
for
z ∈ ∂Ω
with gˆj := ⟨g , sj ⟩χ,Σ
(4.9)
j=1
and ∥g ∥2χ ,Σ =
∑∞
ˆ 2 . Define
j=1 gj
∫ uM (x) :=
Σ
NM (x, z) g(z) χ dσ
with NM (x, z) :=
M ∑ sj (x) sj (z). λj
(4.10)
j=1
Note that the range of this operator is the subspace of V0 spanned by the first M mS-eigenfunctions of this problem. The following theorem provides a bound on the solution in terms of the boundary data. Theorem 4.1. Assume (B1)–(B3) hold and the sequence {uM : M ≥ 1} is defined by (4.10). Then the uM converge strongly in 1 HΣ (Ω ) to the unique solution u˜ of (4.1) and
∫ Ω
χ |∇ u˜ |2 dx =
∫ ∞ 2 ∑ 1 gˆj g 2 χ dσ . ≤ λj λ1 Σ
(4.11)
j=1
1 Proof. ∑∞ The solution u˜ of Eq. (4.1) is χ -orthogonal to H0 (Ω ) so it lies−1in V0 . From (R3), it has the representation u˜ (x) = c s (x). Substitute this in (4.1) and take v = s , to see that cj = λj gˆj for each j ≥ 1. Thus the solution is j k=1 k k
u˜ (x) =
∞ ∑ gˆj sj (x). λj j=1
1 The uM of (4.10) is just the Mth partial sum of this series so they converge to u˜ strongly in HΣ (Ω ) from the Riesz–Fisher theorem. The equality in (4.11) holds from the formulae for the eigenfunctions. Since the λj are increasing the inequality in (4.11) follows. □
This result is somewhat surprising as the original equation (1.1) involves a non-self-adjoint operator so one ordinarily does not expect representations of solutions in terms of orthogonal eigenfunction expansions. These Steklov representation of solutions of problems with inhomogeneous boundary conditions provide computable approximations for problems that historically have generally been treated by boundary integral operators and equations. Acknowledgment This research was supported in part by NSF award DMS 11008754. References [1] A.N. Brooks, T.J.R. Hughes, Streamline upwind Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982) 199–259. [2] R. Glowinski, Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems, SIAM, Philadelphia, 2005. [3] H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces, SIAM Publications, Philadelphia, 2006. [4] M. Ortiz, A variational formulation for convection-diffusion problems, Internat. J. Engrg. Sci. 23 (1983) 717–731. [5] A. Ambrosetti, G. Prodi, A Primer of Nonlinear Analysis, Cambridge U. Press, Cambridge, 1993. [6] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton U. Press, Princeton, 1983. [7] G. Auchmuty, Finite energy solutions of self-adjoint elliptic mixed boundary value problems, Math. Methods Appl. Sci. 33 (2010) 1446–1462. [8] G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim. 25 (2004) 321–348. [9] G. Auchmuty, Bases and comparison results for linear elliptic eigenproblems, J. Math. Anal. Appl. 390 (2012) 394–406.
Please cite this article in press as: G. Auchmuty, Variational principles for advection–diffusion problems, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.09.023.
)
–
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Variational principles for advection–diffusion problems Giles Auchmuty Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA
article
info
Article history: Available online xxxx Keywords: Advection–diffusion equations Mixed boundary conditions Variational principles
a b s t r a c t Variational principles for linear and semilinear advection–diffusion problems with velocity field given by potential flow are described and analyzed. Mixed Dirichlet and prescribed flux conditions are treated. Existence and uniqueness results are proved and equivalent integral operator equations are found. A positive multiplier function related to the potential of the flow is used to change the system to divergence form. The dependence of the solution on inhomogeneous flux boundary data is determined. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction This paper describes variational principles and some results about the solutions of linear and semilinear advection– diffusion equations in bounded regions and subject to mixed Dirichlet and flux boundary conditions. Google searches provide a tremendous number of papers on the numerical simulation of these problems; a paper by Brooks and Hughes [1] has over 5000 citations in Google scholar. These equations arise in some very disparate areas of science and engineering. Most of the papers describe the numerical computation of solutions under different assumptions on the data. There has also been extensive study of algorithms and procedures for efficiently simulating the observed phenomenology. See chapter 5 of Glowinski [2] for a recent discussion of least squares and other algorithms for these systems. The numerical simulations indicate that, when the advection term is significant, it is better to use methods that respect properties of the velocity field. General existence results for these systems are known based on standard elliptic theory. See section 6.9 of Attouch, Buttazzo and Michaille [3] where existence–uniqueness is proved under an assumption of ‘‘slow flow’’ and zero Dirichlet boundary conditions. Some texts on finite element simulations also prove some existence results. However there has been relatively little mathematical analysis of problems with physically interesting non-zero boundary conditions that help understand the results observed when the velocity terms are significant. Here the construction and properties of variational principles for the solutions will be treated for velocity fields that are gradient flows. This boundary value problem is to find solutions of
− ∆u + a · ∇ u = f (x, u) on Ω
(1.1)
subject to mixed Dirichlet and flux boundary conditions u=0
on
˜ and Dν u = g(x) Σ
on Σ .
(1.2)
Here a : Ω → Rn is assumed to be a gradient field on the bounded region Ω ⊂ Rn with Lipschitz boundary ∂ Ω . Σ is a ˜ = ∂ Ω \ Σ , ν is the unit outward normal at a point on the boundary and f , g are proper open subset of the boundary, Σ prescribed functions on Ω × R, Σ respectively. Further conditions on these functions will be specified below. E-mail address: [email protected]. https://doi.org/10.1016/j.camwa.2017.09.023 0898-1221/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: G. Auchmuty, Variational principles for advection–diffusion problems, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.09.023.
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(1.1)–(1.2) are of the form treated by Ortiz [4] who described some quite different variational principles for this problem. Note that if a is a non-zero vector field, then (1.1) is not in divergence form, nor is it a potential equation, that holds at the critical points of a Gateaux-differentiable function on a Sobolev space. Nevertheless, under the condition that a = − ∇ ϕ is a gradient field, there is a functional E on a natural Hilbert Sobolev 1 (Ω ) whose critical points are precisely the solutions of this system. This functional involves a multiplier (or space HΣ integrating factor) determined by the potential of the flow field a. It appears that this multiplier plays a role analogous to that of preconditioners for the numerical simulation of solutions. The variational principle is used to prove an existence–uniqueness result for the system subject to simple conditions on the source (or reaction) term and standard assumptions on the region and the boundary conditions. The linear case is described carefully in Section 3 and an equivalent integral formulation is outlined. Section 4 provides representations and approximations of solutions that satisfy the inhomogeneous boundary data. These use a class of mixed Steklov eigenfunctions for the problem. 2. Assumptions and spaces For the most part, the notation is standard and similar to that used in [3]. The assumptions on the region Ω , its boundary ˜ are appropriate for the use of finite element models. ∂ Ω and the subsets Σ , Σ (B1) Ω is a bounded connected open set in Rn whose boundary ∂ Ω is the union of a finite number of disjoint closed Lipschitz surfaces; each surface having finite surface area and a unit outward normal ν (.) defined σ a.e.. (When n = 2, these are Lipschitz curves of finite length). ˜ have strictly positive surface area, σ (∂ Σ ) = 0 and g ∈ LqT (Σ , dσ ) with (B2) Σ is a nonempty open subset of ∂ Ω , Σ and Σ qT = 2(1 − 1/n) for n ≥ 3 (qT > 1 when n = 2). The advection term is taken to be a gradient field a(x) := − ∇ ϕ (x) with potential ϕ . A necessary condition for this is that curl a ≡ 0 on Ω when n = 2 or 3. The potential ϕ should satisfy (B3) ϕ is a Lipschitz (W 1,∞ ) function with ϕ (x) ≥ 0 on Ω . Observe that ideal fluid flows are potential flows and the positivity requirement on ϕ holds without loss of generality since adding a constant to a potential function does not change the gradient. When a(x) ≡ a is a constant ∑n field2 the potential has the form ϕ (x) = a · x + a0 . When a(x) = 2 c · x + c0 is linear, the potential is quadratic; ϕ (x) = j=1 cj xj + c0 · x + a0 with a0 chosen so ensure the positivity of ϕ on the compact set Ω . 1 Let Γ : H 1 (Ω ) → L2 (∂ Ω ) be the boundary trace operator and HΣ (Ω ) be the subspace of H 1 (Ω ) of all functions that ˜ . This is a closed subspace of H 1 (Ω ) and will be a real Hilbert space with respect to the inner satisfy Γ u(x) = 0 σ a.e. on Σ product
⟨u, v⟩ :=
∫
∇ u · ∇ v dx
Ω
since Σ is non-empty.
1 A weak formulation of the problem is to find u ∈ HΣ (Ω ) that satisfies
a(u, v ) :=
∫ Ω
[ ∇ u · ∇ v + (a · ∇ u ) v ] dx = m(u, v ) for all v ∈ HΣ1 (Ω ).
(2.1)
Here dx denotes Lebesgue measure and m(u, v ) :=
∫ Ω
f (x, u) v dx +
∫ Σ
g(x) v dσ
(2.2)
with dσ being surface area measure on the boundary ∂ Ω . Define χ (x) := eϕ (x) then χ ∈ W 1,∞ (Ω ) and χ (x) ≥ 1 on Ω ; χ will be called a multiplier for this variational problem. It is easily verified that u solves (1.1) provided
− div( χ ∇ u ) = χ (x) f (x, u) on Ω .
(2.3)
This equation is in divergence form so there is a variational principle for its solutions. Consider the problem (P ) of minimizing 1 1 E on HΣ (Ω ) where E : HΣ (Ω ) → R is defined by
∫
χ | ∇ u | − 2 F (x, u) dx − 2
[
E (u) := Ω
2
]
∫ Σ
χ g u dσ
(2.4)
∫s
with F (x, s) := 0 f (x, t) dt. Note that this functional involves the advection field solely through the multiplier χ (x). To obtain existence results for this variational problem we require the following condition on the source term f and its antiderivative F. (B4) Assume f (., s) is Borel measurable on Ω for each s ∈ R, f (x, .) is continuous on R for almost all x ∈ Ω and for each ϵ > 0 there is a C (ϵ ) such that F (x, s) ≤ C (ϵ ) + ϵ s2 on Ω × R. This is the condition that f is a Caratheodory function whose indefinite integral is sub-quadratic in s. In particular it holds when f is bounded and continuous on Ω × R. Please cite this article in press as: G. Auchmuty, Variational principles for advection–diffusion problems, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.09.023.
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1 Theorem 2.1. Assume (B1)–(B4) hold, then E is weakly lower semi-continuous (w.l.s.c) and coercive on HΣ (Ω ), and there are 1 minimizers of E on H (Ω ). The minimizers satisfy
∫ Ω
χ [ ∇ u · ∇ v − f (x, u) v ] dx −
∫ Σ
χ g v dσ = 0 for all v ∈ HΣ1 (Ω ).
(2.5)
1 (Ω ) so they are w.l.s.c. To prove Proof. The first and last terms in the expression for E are convex and continuous on HΣ 1 that second term is w.l.s.c. let uk be a sequence of functions that converges weakly to a limit u˜ in HΣ (Ω ). Then uk converges 2 1 strongly to u˜ in L (Ω ) from Rellich’s theorem and thus in L (Ω ) as Ω is bounded. Decompose F (x, s) = F+ (x, s) − F− (x, s) into its ∫ positive and negative parts ∫with F+ (x, s) := max(F (x, s), 0). From theorem 2.3 of Ambrosetti and Prodi, [5], it follows that Ω F+ (x, uk ) dx converges∫ to Ω F+ (x, u˜ ) dx as k → ∞ ∫ in view of the upper bound requirement of (B4). Theorem 2.3 of Giaquinta [6] implies that Ω F− (x, u˜ ) ≤ lim infk→∞ Ω F− (x, uk ) dx so this functional is w.l.s.c. Thus each term in the expression for E is w.l.s.c. ∫ ∫ ˜ has positive surface measure there is a c0 > 0 such that Since Ω is bounded and Σ | ∇ u |2 dx ≥ c0 Ω u2 dx for all Ω 1 1 (Ω ) and there will be a minimizer of E as claimed. (Ω ). Choose ϵ < c0 in (B4), then the functional E is coercive on HΣ u ∈ HΣ 1 1 When v ∈ HΣ (Ω ) ∩ C (Ω ), the first variation of E at u ∈ HΣ (Ω ) is δ E (u, v ) = limt →0 t −1 [ E (u + t v ) − E (u) ] . This exists 1 and is given by the left hand side of (2.5). At the minimizer it will be zero, so (2.5) holds by density for all v ∈ HΣ (Ω ). □
Observe that (2.5) is a weak version of Eq. (2.3) so the minimizers of E on H 1 (Ω ) are weak solutions of (2.3)–(1.2). The next section investigates the case where f (x, s) is an affine function in more detail. 3. Solution operators for linear advection–diffusion Here some properties of the solutions of this problem in the case where the original system (1.1) is linear will be described. Consider the case where the equation is
− ∆u − ∇ ϕ · ∇ u + c u = f0 (x)
on Ω .
(3.1)
The analog of Eq. (2.3) now is
− div( χ ∇ u ) + χ c u = χ (x) f0 (x)
on
Ω.
(3.2)
Consider the problem of minimizing E (.) on HΣ (Ω ) with E (u) = E1 (u) + F (u) where 1
∫ E1 (u) :=
χ | ∇ u | dx, 2
Ω
∫
χ c u − 2 f0 u dx − 2
[
F (u) := Ω
2
]
∫ Σ
χ g u dσ .
(3.3)
Note that this differs from the energy functions for standard Poisson type problems by the inclusion of the weight function χ := eϕ (x) ≥ 1 in each integral. To prove existence and uniqueness the following assumptions on f0 , c are imposed. (B4L) Assume f0 , c are Borel measurable functions with c ≥ 0 in Ln/2 (Ω ), f0 ∈ LqS (Ω ), qS = 2n/(n + 2) when n ≥ 3, or else qS > 1 when n = 2. This variational problem now has the following properties — so its minimizer is a weak solution of (3.2). Theorem 3.1. Assume (B1)–(B3) and (B4L) hold, then F is weakly lower semicontinuous and the functional E is strictly convex, 1 1 continuous and coercive on HΣ (Ω ). The unique minimizer of E is the unique solution in HΣ (Ω ) of
∫ Ω
χ [ ∇ u · ∇ v + ( c u − f0 )v ] dx −
∫ Σ
χ g v dσ = 0 for all v ∈ HΣ1 (Ω ).
(3.4)
Proof. When (B4L) holds, then the Sobolev imbedding theorem implies that each term in F is well-defined and continuous 1 on HΣ (Ω ). The quadratic term is convex as c ≥ 0 so the functional E is coercive and strictly convex. Hence the functional attains a unique minimizer. The proof that this minimizer satisfies (3.2) is similar to that of Theorem 2.1. □ 1 Observe that (3.4) is a weak formulation of (3.2). When g ≡ 0 let G (f0 ) be the minimizer of this E on HΣ (Ω ) and the minimizer when f ≡ 0 be denoted Gb (g). Consider the restriction of these solution operators to data f0 , g in L2 (Ω ), L2 (∂ Ω ) respectively. 1 Theorem 3.2. Assume that (B1)–(B3) and (B4L) hold, then the maps G , Gb are continuous linear maps into HΣ (Ω ). They are compact linear transformations of L2 (Ω ), L2 (∂ Ω ) respectively into L2 (Ω ). 1 Proof. The verification that G , Gb are linear transformations is standard. They are bounded maps from LqS (Ω ) to HΣ (Ω ) 1 qT and from L (Σ ) to HΣ (Ω ) with the choice of qS , qT from conditions (B2) and (B4L) upon using the Sobolev imbedding and 1 trace theorems and coercivity inequalities. Hence they are continuous linear maps of G , Gb respectively into HΣ (Ω ). From 1 2 Rellich’s theorem, the imbedding of HΣ (Ω ) into L (Ω ) is compact, so by composition both G , Gb will be compact maps from L2 (Ω ), L2 (∂ Ω ) since qS < 2 and qT < 2 for all n ≥ 2. □
Please cite this article in press as: G. Auchmuty, Variational principles for advection–diffusion problems, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.09.023.
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Since these are compact linear transformations, they will have good finite rank approximations and may be written as
∫ (G f0 )(x) :=
G(x, y) f0 (y) dy
Ω
∫ and
(Gb g)(x) :=
Σ
Gb (x, z) g(z) dσ (z)
(3.5)
where the integral kernels G, Gb are limits of appropriate Galerkin approximations. In particular the kernel G can be approximated using eigenfunctions and eigenvalues of the operator Lu := χ c u − div(χ ∇ u ) subject to zero-Dirichlet ˜ and zero-Neumann conditions on Σ . boundary conditions on Σ The solutions of the system (3.3) can now be shown to be the solutions u ∈ L2 (Ω ) of the linear compact operator equation
∫ u(x) = Σ
Gb (x, z) g(z) dσ (z) +
∫ Ω
G(x, y) f0 (y) dy.
(3.6)
This equation is of a form for which there is an extensive analytic and computational theory. In particular it may be used to study the dependence of the solutions on the interior sources f and/or the boundary data g through the subset Σ of the boundary ∂ Ω . 4. Dependence of solutions on the boundary flux Physically, an important question for these problems is the dependence of the solutions on the boundary flux g. From (B2) and Theorem 2.1, there will be an H 1 -solution of these problems whenever g ∈ L4/3 (Σ ) for problems posed on regions in R3 . In this section, an explicit representation of a solution of the homogeneous equation div(χ ∇ u ) = 0 on Ω subject to the boundary conditions (1.2) will be described. The expressions are obtained via similar methods to those described in [7] for some different mixed boundary value problems. The boundary conditions associated with the minimizers of E defined by (2.4) are weak forms of (1.2). That is we seek 1 properties of solutions u˜ ∈ HΣ (Ω ) of the system
∫ Ω
χ ∇ u · ∇ v dx =
∫ Σ
g v χ dσ
for all
v ∈ HΣ1 (Ω ).
(4.1)
Some different representations and formulae for this solution may be found depending on how terms involving the function c are treated. A choice that involves the velocity potential ϕ , but is independent of the source term, may be found by using mixed 1 Steklov (mS-)eigenfunctions that produce orthogonal bases of HΣ (Ω ) with respect to the inner product
⟨u, v⟩χ :=
∫
χ ∇ u · ∇ v dx +
Ω
∫ Σ
u v χ dσ .
(4.2)
1 1 This is called the χ -inner product on HΣ (Ω ) and it is straightforward to verify that HΣ (Ω ) is a real Hilbert space 1 with respect to this inner product. A function s ∈ HΣ (Ω ) is a mS-eigenfunction corresponding to an eigenvalue λ provided it is a non-zero solution of the system
a(u, v ) :=
∫ Ω
χ ∇ s · ∇ v dx = λ
∫ Σ
χ s v dσ
for all
v ∈ HΣ1 (Ω ).
(4.3)
Such functions s are weak solutions of the system div( χ∇ s ) = 0
on Ω
with s = 0
on
˜, Σ
Dν s = λ s
on
Σ.
(4.4)
1 Let V0 to be the subspace of HΣ (Ω ) of functions u that satisfy
a(u, v ) =
∫ Ω
χ ∇ u · ∇ v dx = 0 for all v ∈ H01 (Ω ).
(4.5)
These functions are weak solutions of the differential equation in (4.4) so V0 may be regarded as the subspace of χ -harmonic 1 functions in HΣ (Ω ). Then 1 HΣ (Ω ) = H01 (Ω )⊕χ V0
(4.6)
with ⊕χ denoting orthogonal sum with respect to the χ -inner product (4.2). This has a similar form to those studied first in [8]. The mS-eigenproblem is of the type studied in Auchmuty ∫ [9] and the eigenfunctions satisfying (4.3) are in V0 . In the 1 notation of [9], a is as above while V = HΣ (Ω ), m(u, v ) := Σ χ u v dσ . Theorem 3.1 then yields that there is a least, strictly 1 positive, mS-eigenvalue λ1 of (4.3) and a corresponding mS-eigenvalue s1 ∈ HΣ (Ω ). Moreover the coercivity inequality
∫
χ | ∇ u | dx ≥ λ1 2
Ω
∫ Σ
χ u2 d σ
(4.7)
Please cite this article in press as: G. Auchmuty, Variational principles for advection–diffusion problems, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.09.023.
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1 holds for all u ∈ HΣ (Ω ). Let L2 (Σ , χ dσ ) be the usual Lebesgue space with the inner product
⟨u, v⟩χ ,Σ :=
∫ Σ
u v χ dσ .
(4.8)
Results from section 4 of [9] show that there is an increasing sequence Λ := {λj : j ≥ 1} of mS-eigenvalues and associated mS-eigenfunctions S := {sj : j ≥ 1} with (R1) λj → ∞ as j → ∞, (R2) the boundary traces of the functions in S have support in Σ and are an orthonormal basis of L2 (Σ , χ dσ ). (R3) S is a maximal χ -orthogonal set in V0 . In view of (R2), elementary Hilbert space theory says that g ∈ L2 (Σ , χ dσ ) has a Fourier expansion of the form g(z) =
∞ ∑
gˆj sj (z)
for
z ∈ ∂Ω
with gˆj := ⟨g , sj ⟩χ,Σ
(4.9)
j=1
and ∥g ∥2χ ,Σ =
∑∞
ˆ 2 . Define
j=1 gj
∫ uM (x) :=
Σ
NM (x, z) g(z) χ dσ
with NM (x, z) :=
M ∑ sj (x) sj (z). λj
(4.10)
j=1
Note that the range of this operator is the subspace of V0 spanned by the first M mS-eigenfunctions of this problem. The following theorem provides a bound on the solution in terms of the boundary data. Theorem 4.1. Assume (B1)–(B3) hold and the sequence {uM : M ≥ 1} is defined by (4.10). Then the uM converge strongly in 1 HΣ (Ω ) to the unique solution u˜ of (4.1) and
∫ Ω
χ |∇ u˜ |2 dx =
∫ ∞ 2 ∑ 1 gˆj g 2 χ dσ . ≤ λj λ1 Σ
(4.11)
j=1
1 Proof. ∑∞ The solution u˜ of Eq. (4.1) is χ -orthogonal to H0 (Ω ) so it lies−1in V0 . From (R3), it has the representation u˜ (x) = c s (x). Substitute this in (4.1) and take v = s , to see that cj = λj gˆj for each j ≥ 1. Thus the solution is j k=1 k k
u˜ (x) =
∞ ∑ gˆj sj (x). λj j=1
1 The uM of (4.10) is just the Mth partial sum of this series so they converge to u˜ strongly in HΣ (Ω ) from the Riesz–Fisher theorem. The equality in (4.11) holds from the formulae for the eigenfunctions. Since the λj are increasing the inequality in (4.11) follows. □
This result is somewhat surprising as the original equation (1.1) involves a non-self-adjoint operator so one ordinarily does not expect representations of solutions in terms of orthogonal eigenfunction expansions. These Steklov representation of solutions of problems with inhomogeneous boundary conditions provide computable approximations for problems that historically have generally been treated by boundary integral operators and equations. Acknowledgment This research was supported in part by NSF award DMS 11008754. References [1] A.N. Brooks, T.J.R. Hughes, Streamline upwind Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982) 199–259. [2] R. Glowinski, Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems, SIAM, Philadelphia, 2005. [3] H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces, SIAM Publications, Philadelphia, 2006. [4] M. Ortiz, A variational formulation for convection-diffusion problems, Internat. J. Engrg. Sci. 23 (1983) 717–731. [5] A. Ambrosetti, G. Prodi, A Primer of Nonlinear Analysis, Cambridge U. Press, Cambridge, 1993. [6] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton U. Press, Princeton, 1983. [7] G. Auchmuty, Finite energy solutions of self-adjoint elliptic mixed boundary value problems, Math. Methods Appl. Sci. 33 (2010) 1446–1462. [8] G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim. 25 (2004) 321–348. [9] G. Auchmuty, Bases and comparison results for linear elliptic eigenproblems, J. Math. Anal. Appl. 390 (2012) 394–406.
Please cite this article in press as: G. Auchmuty, Variational principles for advection–diffusion problems, Computers and Mathematics with Applications (2017), https://doi.org/10.1016/j.camwa.2017.09.023.