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Mathematical Models and Methods in Applied Sciences
c World Scientific Publishing Company
Subgrid modeling for convection-diffusion-reaction in 2 space
dimensions using a Haar Multiresolution analysis
Johan Hoffman
Courant Institute, New York University, 251 Mercer Street
New York, New York 10012-1185, USA
[email protected]
Received (Day Month Year)
Revised (Day Month Year)
Communicated by (xxxxxxxxxx)
In this paper we study a subgrid model based on extrapolation of a modeling residual,
in the case of a linear convection-diffusion-reaction problem Lu = f in two dimensions.
The solution u to the exact problem satisfies an equation Lh u = [f ]h + Fh (u), where Lh
is the operator used in the computation on the finest computational scale h, [f ]h is the
approximation of f on the scale h, and Fh (u) is a modeling residual, which needs to be
modeled. The subgrid modeling problem is to compute approximations of Fh (u) without
using finer scales than h. In this study we model Fh (u) by extrapolation from coarser
scales than h, where Fh (u) is directly computed with the finest scale h as reference.
We show in experiments that a solution with subgrid model on a scale h in most cases
corresponds to a solution without subgrid model on a mesh of size less than h/4.
Keywords: dynamic subgrid modeling, Haar multiresolution analysis, convectiondiffusion-reaction, scale extrapolation
AMS Subject Classification: 65M60,76F25, 76F65
1. Introduction
A fundamental problem in science and engineering concerns the mathematical modeling of phenomena involving small scales. This problem arises in molecular dynamics, turbulent flow and flow in heterogeneous porous media, for example. Basic
models for such phenomena, such as the Schrödinger equation or the Navier-Stokes
equations, may be very accurate models of the real phenomena but may be so computationally intensive, because of the large number of degrees of freedom needed
to represent the small scales, that even computers with power way beyond that
presently available may be insufficient for accurate numerical solutions of the given
equations. The traditional approach to get around this difficulty is to seek to find
simplified models with computationally resolvable scales, whose solutions are sufficiently close to the solutions of the original full equations. Such simplified models,
without the too small scales, build on mathematical modeling of the computationally unresolved scales of the full equations, which is referred to as subgrid modeling.
1
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To find suitable simplified models, including subgrid modeling, is the central activity in modeling of turbulence, molecular dynamics and heterogeneous media. For
references on subgrid modeling in turbulence, we refer to Gatski et.al. 5 and Sagaut
15
.
In recent years, new approaches to the subgrid modeling problem have been
taken based on dynamic subgrid modeling, an idea first introduced by Germano et
al. 6 . The basic assumption here is that a particular model applies on different scales
with the same value on the model parameters. Using this assumption, one seeks
to find a subgrid model, for each set of data, by computing approximations of the
subgrid model on coarser scales using a fine scale computed solution without subgrid
model as reference, and then finally extrapolating the so obtained model to the
finest computational scale, with the hope of being able to extrapolate from the finest
resolvable scales to unresolvable scales. In order for such a dynamic modeling process
based on extrapolation to work, it is necessary that the underlying problem has some
“scale similarity”, so that the experience gained by fitting the model on a coarse
scale with a fine scale solution as reference, may be extrapolated to the finer scale.
It is conceivable that many problems involving a range of scales from large to small,
such as fluid flow at larger Reynolds numbers, in fact does have such a regularity,
once the larger scales related to the geometry of the particular problem have been
resolved. The purpose of this note is to study the feasibility of the indicated dynamic
subgrid modeling in the context of some simple model problems related to linear
convection-diffusion-reaction with irregular or non-smooth coefficients with features
on many scales. The scale similarity in this case appears to be close to assuming that
the coefficients have a “fractal nature” and that the solution inherits this structure
to some degree.
The problem of computational mathematical modeling has two basic aspects:
numerical computation and modeling. The basic idea in dynamic subgrid modeling
is to seek to extrapolate into unresolvable scales by comparing averaged fine scale
computed solutions of the original model (without subgrid modeling) on different
coarser scales. To make the extrapolation possible at all, the numerical errors in the
computations underlying the extrapolation have to be small enough. If the numerical errors in the fine scale computation without subgrid model are not sufficiently
small, then the whole extrapolation procedure from coarser scales may be meaningless. Thus, it will be of central importance to accurately balance the errors from
numerical computation and subgrid modeling. In recent years the techniques for
adaptive error control based on a posteriori error estimates have been considerably
advanced 3,1,10 , and thus today we have techniques available that allow the desired
balance of computational and modeling errors.
This is the second paper in this series. In the first paper 11 , the feasibility of the
dynamic computational subgrid model was investigated for some very simple one
dimensional model problems related to convection-diffusion-reaction with certain
non smooth “scale similar” coefficients. The subgrid model was based on extrapolation of a modeling residual, and the modeling error was studied separately by
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making the numerical errors neglible by computing on a very fine mesh. A priori
and a posteriori error estimates for the modeling errors were also presented.
The first part of is paper is again focused on modeling only. We then introduce
numerical errors, and the relation between the numerical and the modeling errors is
studied. The theoretical results of the first paper 11 are extended to two dimensions,
and a posteriori error estimates in terms of one modeling residual and one numerical
residual are presented. In a continuation of this study non linear unsteady problems
are studied 7 , and support for the scale similarity assumption used in this paper is
presented 10 . Extensions to 3d Navier-Stokes equations connecting to LES is under
way 8 .
An outline of this note is as follows: In Section 2 we introduce the linear
convection-diffusion-reaction model problem, and we recall the basic results of the
first paper 11 . In particular, the idea of a subgrid model based on extrapolation
of a modeling residual is presented. In Section 3 we recall basic features of a Multiresolution Analysis MRA using the two dimensional Haar basis, which we use to
motivate an Ansatz on the form of the modeling residual, which we then base the
extrapolation upon. In Section 4 we present error estimates, and in Section 5 we test
the method in the context of some numerical experiments in two dimensions with
non smooth “scale similar” coefficients. We first consider the modeling errors separately by computing on a very fine mesh that makes the numerical errors neglible
compared to the modeling errors. We then introduce numerical errors by computing
on a mesh corresponding to the finest resolvable scale in the modeling problem. We
conclude with some remarks in Section 6.
2. Formulation of the Problem
As a model problem we consider the scalar convection-diffusion-reaction problem
of the form
Lu(x) = −∇ · (ǫ(x)∇u(x)) + β(x) · ∇u(x) + α(x)u(x) = f (x),
(2.1)
for x ∈ Ω ⊂ Rn , together with boundary conditions for the inflow and outflow
boundaries ∂Ω− = {x ∈ ∂Ω : β(x) · n(x) < 0} and ∂Ω+ = {x ∈ ∂Ω : β(x) · n(x) >
0}, where n is an unit outward normal from Ω. ǫ(x), β(x), and α(x) are given
coefficients depending on x, f (x) is a given force, and u(x) is the solution. We
assume that the coefficients ǫ, β, and α are piecewise continuous, and we seek a
weak solution u that satisfies the corresponding variational formulation of (2.1),
and the boundary conditions in the sense of traces.
We assume that the coefficients ǫ, β and α, and the given function f vary on a
range of scales from very fine to coarse scales, and we expect the exact solution u
in general to vary on a related range of scales. We denote by h the finest possible
scale we allow us to use, which may be the finest possible scale in a computation of
a solution, and we denote the corresponding approximate solution uh . We assume
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for now that uh is the weak solution of the following simplified problem
Lh uh = −∇ · ([ǫ]h ∇uh ) + [β]h · ∇uh + [α]h uh = [f ]h ,
x ∈ Ω,
(2.2)
together with boundary conditions, where [ǫ]h , [β]h , [α]h , and [f ]h are approximations of the corresponding functions on the scale h, with the finer scales left out.
We may think of uh as an approximation of the exact solution u obtained by simplifying the model by simplifying the coefficients in the model removing scales finer
than h. Typically, the coefficient [β]h is some local average of β on the scale h, etc.
The difference u − uh thus represents a modeling error resulting from averaging the
coefficients on the scale h.
2.1. Dynamic subgrid modeling
We now consider a situation where uh is not a sufficiently good approximation of
u, and we would like to improve the quality of uh without computing using finer
scales than h. The equation Lu = f satisfied by the exact solution can be written
in the form
Lh u = [f ]h + Fh (u),
(2.3)
Fh (u) = f − [f ]h − (L − Lh )u
(2.4)
where
acts as a modeling residual. The subgrid modeling problem is to model Fh (u) on
the scale h. There is a variety of possibilities to approach this problem. We may use
Fh (u) as a correction on the force and replace the model Lh uh = [f ]h by the model
Lh ũh = [f ]h + F̃h ,
(2.5)
with solution ũh , where F̃h is an approximation of Fh (u). Alternatively, we may
seek to model L − Lh as a correction L̃h of the operator Lh and solve a modified
problem of the form
(Lh + L̃h )ũh = f,
(2.6)
where thus the correction L̃h acts as a model of L − Lh . In the first approach the
subgrid model takes the form of a corrective force F̃h independent of ũh , and in the
second approach the subgrid model also contains a correction L̃h ũh depending on
ũh .
A method for computing an approximation of the modeling residual Fh (u) in
(2.3) was proposed by Hoffman et.al. 11 using the idea of a dynamic model, in the
case of a linear convection-diffusion-reaction problem. It was shown that the local
average of Fh (u) on the scale h is equal to a sum of covariances of the form [vw]h −
[v]h [w]h . Based on a Haar MRA, an Ansatz of the form [vw]h − [v]h [w]h ≈ Chµ was
proposed. The two functions C(x) and µ(x) were determined by extrapolation from
computing approximations FH (uh ) of the modeling residual FH (u) on two coarser
scales H, where the solution to the simplified problem uh was used as a substitute
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for the solution u to the exact problem. A priori and a posteriori modeling error
estimates for the one dimensional case were presented. Numerical experiments in
one dimension were also presented, indicating that the modeling error in a corrected
solution ũh on the scale h is less than the modeling error in a non corrected solution
on the scale h/4. Only modeling errors were considered, and the numerical errors
were made neglible by computing on a very fine computational mesh.
3. Scale Extrapolation using MRA
The notion of a MRA was introduced in the early 90’s by Meyer 14 and Mallat 13
as a general framework for construction of wavelet bases. Hoffman et.al. 11 used a
Haar MRA in L2 ([0, 1]) to motivate a subgrid model based on extrapolation of the
modeling residual Fh (u). In this section we extend the results of Hoffman et.al. 11
to two space dimensions, using the Haar basis in L2 ([0, 1]2 ). Based on these results
we formulate an Ansatz on Fh (u), which we use to extrapolate an approximation
F̃h to Fh (u).
3.1. MRA
An orthonormal MRA of L2 ([0, 1]) is a decomposition of L2 ([0, 1]) into a chain of
closed subspaces
V0 ⊂ V1 ⊂ ... ⊂ Vj ⊂ ...
such that
[
Vj = L2 ([0, 1]).
j≥0
Each Vj is spanned by the dilates and integer translates of one scale function ϕ ∈ V0 :
Vj = span{ϕj,k (x) = 2j/2 ϕ(2j x − k)},
and the functions ϕj,k form an L2 -orthonormal basis in Vj . We denote the orthogonal
complement of Vj in Vj+1 by Wj , which is generated by another orthonormal basis
(the wavelets) ψj,k (x) = 2j/2 ψ(2j x − k), where ψ ∈ W0 is called the mother wavelet.
The space L2 ([0, 1]) can now be represented as a direct sum
L2 ([0, 1]) = V0 ⊕ W0 ⊕ ... ⊕ Wj ⊕ ...
For a more detailed presentation of the MRA concept we refer to Daubechies
Louis et.al. 12 .
2
or
3.2. The Haar MRA in L2 ([0, 1]2 )
In the case of the two dimensional Haar basis in L2 ([0, 1]2 ), the space V0 is spanned
by the scale function
Φ(x, y) = ϕ(x)ϕ(y),
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−1
+1
−1
+1
−1
−1
+1
+1
+1
−1
+1
−1
ΨH (x, y)
ΨV (x, y)
ΨD (x, y)
and Vj = span{Φj,k (x, y) = 2j Φ(2j x − kx , 2j y − ky )}. The orthogonal complement
of Vj in Vj+1 is Wj = WjH ⊕ WjV ⊕ WjD , where WjH , WjV and WJD are spanned by
the wavelets
j H i
i
ΨH
i,k (x, y) = 2 Ψ (2 x − kx , 2 y − ky ),
ΨVi,k (x, y) = 2j ΨV (2i x − kx , 2i y − ky ),
j D i
i
ΨD
i,k (x, y) = 2 Ψ (2 x − kx , 2 y − ky ),
respectively, where j ∈ Z, k = (kx , ky ) ∈ Z 2 and
ΨH (x, y) = ϕ(x)ψ(y),
ΨV (x, y) = ψ(x)ϕ(y),
ΨD (x, y) = ψ(x)ψ(y).
Here ψ and ϕ are the one dimensional Haar mother wavelet and scale function
respectively, defined by

 1 0 < x < 1/2,
1 0 < x < 1,
ψ(x) = −1 1/2 < x < 1, ϕ(x) =

0 otherwise.
0 otherwise,
Each f ∈ L2 ([0, 1]2 ) has a unique decomposition
X
X
H H
V
D D
(fiH + fiV + fiD ),
f = fΦ Φ +
(fi,k
Ψi,k + fi,k
ΨVi,k + fi,k
Ψi,k ) = fΦ +
i,k
i
where fiH , fiV and fiD represent the contribution on the different scales 2−i corresponding to dyadic subdivisions Si of Ω with mesh points xi,k = kx 2−i and
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yi,k = ky 2−i where kx , ky = 0, 1, ..., 2i , and subdomains si,k = {kx 2−i < x <
ν
(kx + 1)2−i , ky 2−i < y < (ky + 1)2−i }. The coefficients fi,k
(ν = H, V, D) are given
as the L2 inner product of the function f and the corresponding Haar basis function:
Z
ν
fi,k
=
f (x, y)Ψνi,k (x, y) dx dy,
Ω
and fΦ =
R
Ω
f (x, y) dx dy.
3.3. Analysis of Fh (u) using a Haar MRA
In this section we formulate an Ansatz for the modeling residual Fh (u), using the
theory of MRA presented in the previous sections. For f ∈ L2 (Ω), where Ω = [0, 1]2 ,
we define [f ]h to be the piecewise constant function on Si , given by
X
(fjH + fjV + fjD ),
[f ]h = fΦ +
j<i
−i
where we let h = 2 in the rest of this paper. Further, we recall the definition of
the running average f h of a function f ∈ L2 (Ω) on the scale h as
Z x+h/2 Z y+h/2
f h (x, y) = 22i
f (s, t) ds dt,
x−h/2
y−h/2
where (x, y) ∈ Ω and we extend f smoothly outside Ω. We denote by f¯h the
piecewise constant function on the scale h which coincides with f h at the midpoints
of the subdomains si,k . Clearly f¯h is independent of the extension of f outside Ω.
We shall use the following lemma:
Lemma 3.1. f ∈ L2 (Ω) ⇒ [f ]h = f¯h .
Proof. We have
(fjH + fjV + fjD ) ⇒ f¯h = f¯Φh +
f = fΦ +
X
= f¯Φh +
X
j
j<i
h
(f¯jH + f¯jV + f¯jD ) = fΦ +
X
X
h
(fjH + fjV + fjD )
j
(fjH + fjV + fjD ) = [f ]h .
j<i
We recall that Vi is the space of piecewise constant functions on Si , and the
linear mapping L2 ∋ f → [f ]h ∈ Vi can be identified with the L2 -projection of f
onto Vi . Assuming for the moment that ǫ is constant, so that [ǫ]h = ǫ, the modeling
residual is given by
Fh (u) = f − [f ]h − (β · ∇u + αu − [β]h · ∇u − [α]h u).
From the definition we have that [[f ]H g]h = [f ]H [g]h whenever H ≥ h (H = 2−j ,
h = 2−i with j < i). This gives that
[Fh (u)]h = [β]h · [∇u]h − [β · ∇u]h + [α]h [u]h − [αu]h .
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We denote the projection [Fh (u)]h of Fh (u) onto Vi by F̄h (u). We shall now seek to
extrapolate F̄h (u), and we are thus led to study in particular quantities of the form
Eh (v, w) = [vw]h − [v]h [w]h ,
(3.18)
for given functions v and w, which has the form of a covariance. Using the Haar
basis, the covariance Eh (v, w) takes a simple form:
Lemma 3.2. v, w ∈ L2 ⇒ For given (x, y) ∈ Ω,
X
H H
V
V
D D
Eh (v, w)(x, y) =
22j (vj,l
wj,l + vj,l
wj,l
+ vj,l
wj,l ).
j ≥ i
l : x ∈ sj,l
Proof. For v, w ∈ L2 we have
X
X
v = vΦ +
(vjH + vjV + vjD ), w = wΦ +
(wkH + wkV + wkD ),
j
k
and thus
vw = vΦ wΦ + vΦ
X
(wkH + wkV + wkD ) + wΦ
+
(vjH + vjV + vjD )
j
k
X
X
(vjH
+
vjV
+
vjD )(wkH
+
wkV
+ wkD ).
j,k
Similarly
[v]h = vΦ +
X
(vjH + vjV + vjD ), [w]h = wΦ +
j<i
X
(wkH + wkV + wkD ),
k<i
and thus
[v]h [w]h = vΦ wΦ + vΦ
X
(wkH + wkV + wkD ) + wΦ
+
(vjH + vjV + vjD )
j<i
k<i
X
X
(vjH
+
vjV
+
vjD )(wkH
+
wkV
+ wkD ).
j,k<i
Using Lemma 3.1, we obtain
[vw]h = vw h = vΦ wΦ h + vΦ
h
X
(wkH + wkV + wkD )
k
+wΦ
X
h
(vjH + vjV + vjD ) +
j
X
h
(vjH + vjV + vjD )(wkH + wkV + wkD )
j,k
= vΦ wΦ + vΦ
X
(wkH
+
wkV
+ wkD ) + wΦ
j<i
k<i
+
X
(vjH
+
vjV
+
vjD )(wkH
+
wkV
+
wkD )
j,k<i
+
X
j≥i
X
h
(vjH + vjV + vjD )(wjH + wjV + wjD )
(vjH + vjV + vjD )
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= [v]h [w]h +
X
9
(vjH wjH + vjV wjV + vjD wjD ).
j≥i
Finally we have for (x, y) ∈ sj,l that
H H
H H
vjH wjH + vjV wjV + vjD wjD = vj,l
Ψj,l (x, y) wj,l
Ψj,l (x, y)
V
V
D D
D D
+vj,l
ΨVj,l (x, y) wj,l
ΨVj,l (x, y) + vj,l
Ψj,l (x, y) wj,l
Ψj,l (x, y)
D D
H H
V
V
wj,l + vj,l
wj,l
+ vj,l
wj,l ).
= 22j (vj,l
Lemma 3.2 asserts that Eh (v, w) only depends on the scales finer than h,
and that there are no mixing between the scales. Alternatively we can express
Eh (v, w)(x, y) for (x, y) ∈ si,k in terms of all Haar coefficients corresponding to
Haar basis functions with support in si,k , using that Eh (v, w) is constant on si,k
and thus equal to its average over si,k :
Lemma 3.3. v, w ∈ L2 ⇒ For given (x, y) ∈ si,k ,
X
H H
V
V
D D
Eh (v, w)(x, y) = 22i
(vj,l
wj,l + vj,l
wj,l
+ vj,l
wj,l ).
j ≥ i
l : sj,l ⊂ si,k
Proof. From the proof of Lemma 3.2 we have that
X
D D
D D
V
V
H H
H H
Eh (v, w) =
Ψj,l ).
ΨVj,l + vj,l
Ψj,l wj,l
Ψj,l + vj,l
ΨVj,l wj,l
(vj,l
Ψj,l wj,l
j≥i,l
Since Eh (v, w) is constant on si,k and thus equal to its average over si,k we get
Z
Eh (v, w)(x, y) = 22i
Eh (v, w)(s, t) ds dt
si,k
= 22i
=2
2i
Z
si,k j≥i,l
XZ
j≥i
=2
2i
X
si,k
X
D D
D D
V
V
H H
H H
Ψj,l wj,l
Ψj,l ) ds dt
ΨVj,l wj,l
ΨVj,l + vj,l
Ψj,l wj,l
Ψj,l + vj,l
(vj,l
X
H H
H H
V
V
D D
D D
(vj,l
Ψj,l wj,l
Ψj,l + vj,l
ΨVj,l wj,l
ΨVj,l + vj,l
Ψj,l wj,l
Ψj,l ) ds dt
l
j ≥ i
l : sj,l ⊂ si,k
H H
vj,l
wj,l
Z
sj,l
2
D D
(ΨH
j,l ) ds dt + ... + vj,l wj,l
which completes the proof, since
R
ν 2
sj,l (Ψj,l )
Z
sj,l
2
(ΨD
j,l ) ds dt,
ds dt = 1.
An interesting situation is when both v and w are “scale similar” in the sense that
ν
ν
vj,k
= αν 2−j(1+δν ) and wj,k
= βν 2−j(1+γν ) (ν = H, V, D), where αν , βν , δν , γν ∈ Vi ,
ν
ν
which corresponds to v and w having a simple fractal structure (vj+1,l
= 2−δν vj,k
ν
ν
, for sj+1,l ⊂ sj,k ). In that particular situation we find that
and wj+1,l
= 2−γν wj,k
Eh (v, w) has a certain form:
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ν
ν
Corollary 3.1. If v, w ∈ L2 ([0, 1]2 ) with vj,l
= αν 2−j(1+δν ) and wj,l
= βν 2−j(1+γν )
(ν = H, V, D), where αν , βν , δν , γν ∈ Vi , then for (x, y) ∈ Ω
Eh (v, w)(x, y) = CH (x, y)hµH (x,y) + CV (x, y)hµV (x,y) + CD (x, y)hµD (x,y) ,
where Cν = (αν βν )/(1 − 2−(δν +γν ) ) and µν = δν + γν .
Proof. By Lemma 3.2 we have
X
H H
V
V
D D
Eh (v, w)(x, y) =
22j (vj,l
wj,l + vj,l
wj,l
+ vj,l
wj,l )
j ≥ i
l : x ∈ sj,l
=
X
22j (αH 2−j(1+δH ) βH 2−j(1+γH ) + ... + αD 2−j(1+δD ) βD 2−j(1+γD ) )
X
(αH βH 2−j(δH +γH ) + ... + αD βD 2−j(δD +γD ) )
j ≥ i
=
j ≥ i
= αH βH hδH +γH
X
2−(j−i)(δH +γH ) + ... + αD βD hδD +γD
j ≥i
= αH βH hδH +γH
X
2−(j−i)(δD +γD )
j ≥ i
2−j(δH +γH ) + ... + αD βD hδD +γD
j =0
=
X
X
2−j(δD +γD )
j =0
αD βD
αH βH
· hδH +γH + ... +
· hδD +γD
1 − 2−(δH +γH )
1 − 2−(δD +γD )
Assuming that the coefficients in (2.1) are ’scale similar’ in the sense of Corollary
3.1, and that the solution inherits this local fractal structure to some degree, we
have determined the form of the covariance Eh (v, w), and thereby the form of
the modeling residual F̄h (u), and by performing a wavelet transform 2,12 we can
determine the wavelet coefficients of v and w in Eh (v, w). In this paper we are not
going to use the full expression for Eh (v, w), instead we consider a reduced form
with only two unknown parameters. We formulate the following Ansatz: For given
(x, y) ∈ Ω,
Eh (v, w)(x, y) ≈ C(x, y)hµ(x,y) ,
(3.25)
where C, µ ∈ Vi−n , with Ĥ = 2−i+n . The Corollary shows that this is, for example,
the situation when CH ≈ CV ≈ CD and µH ≈ µV ≈ µD , or if the coefficients
in one of the directions ν = H, V, D locally dominates. If Eh (v, w) has this form,
then extrapolation of Eh (v, w) will be possible from knowledge of EH (v, w) and
EĤ (v, w) with h < H < Ĥ, from which the coefficients C(x, y) and µ(x, y) may
be determined. Typically, we will assume that the coefficients have a local fractal
structure. We then expect the solution u to inherit this structure to some degree,
and we expect that extrapolation of the modeling residual F̄h (u) will be possible.
This type of scale similarity with respect to a Haar MRA have, for example, been
found in turbulent aerothermal data ? and in Direct Numerical Simulation DNS 10 .
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4. Error Analysis
We first fix some notation for this section. We denote by L2 (Ω) the Hilbert space
of all real-valued Lebesgue measurable functions defined on Ω, with inner product
and norm defined by
Z
p
(v, w) =
v(x)w(x) dx, kwk = (w, w).
Ω
We define H 1 (Ω) to be the Hilbert space consisting of functions in L2 (Ω) with first
order partial derivatives in L2 (Ω). For simplicity, we further assume that {x ∈ ∂Ω :
[β]h (x) · n(x) < 0} = ∂Ω− and that {x ∈ ∂Ω : [β]h (x) · n(x) > 0} = ∂Ω+ .
The problem Lu = f , with homogeneous boundary conditions v|∂Ω− =
∂v
∂n |∂Ω+ = 0, takes the following variational formulation: Find u ∈ V = {v ∈
H 1 (Ω) : v|∂Ω− = 0} such that
a(u, v) = f (v)
∀v ∈ V,
(4.27)
where the bilinear form a(u, v) is defined by a(u, v) = (ǫ∇u, ∇v) + (β · ∇u, v) +
(αu, v), and f (v) = (f, v). The corresponding simplified problem on the scale h
consists of finding uh ∈ V such that
ah (uh , v) = fh (v)
∀v ∈ V,
(4.28)
where ah (uh , v) = ([ǫ]h ∇uh , ∇v) + ([β]h · ∇uh , v) + ([α]h u, v), and fh (v) = ([f ]h , v).
The solution u to the exact problem (4.27) satisfies
ah (u, v) = Fh (v)
∀v ∈ V,
(4.29)
with Fh (v) = ([f ]h + Fh (u), v) + (Fh∇ (u), ∇v), where
Fh (u) = f − [f ]h − (β · ∇u + αu − [β]h · ∇u − [α]h u),
Fh∇ (u)
h
= [ǫ] ∇u − ǫ∇u,
(4.30)
(4.31)
are modeling residuals. The local averages of Fh (u) and Fh∇ (u) on the scale h are
defined by
F̄h (u) = [β]h · [∇u]h − [β · ∇u]h + [α]h [u]h − [αu]h ,
F̄h∇ (u) = [ǫ]h [∇u]h − [ǫ∇u]h .
We construct approximations (F̃h , F̃h∇ ) of (F̄h (u), F̄h∇ (u)) by extrapolation, and we
get a corresponding corrected solution ũh by solving the problem
ah (ũh , v) = F̃h (v)
where F̃h (v) = ([f ]h + F̃h , v) + (F̃h∇ , ∇v).
∀v ∈ V,
(4.32)
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4.1. Error analysis based on duality
A general approach to obtain a error estimation based on duality is presented by
Eriksson et.al 3 , which would lead us to introduce dual problems corresponding to
(2.1), (2.2), and (2.5). To derive a priori error estimates for eh = u − uh and ẽh =
u − ũh in terms of the modeling residuals (Fh (uh ), Fh∇ (uh )) and the corresponding
approximations (F̃h , F̃h∇ ), we consider a dual problem on the form: Find ϕ ∈ V such
that ah (w, ϕ) = (w, Ψ) for all w ∈ V . We then let w = eh and w = ẽh respectively,
which gives the error estimates
(eh , Ψ) = ah (eh , ϕ) = ah (u, ϕ) − ah (uh , ϕ) = Fh (ϕ) − fh (ϕ)
= (Fh (u), ϕ) + (Fh∇ (u), ∇ϕ) ≤ kFh (u)kkϕk + kFh∇ (u)kk∇ϕk,
(ẽh , Ψ) = ah (ẽh , ϕ) = ah (u, ϕ) − ah (ũh , ϕ) = Fh (ϕ) − F̃h (ϕ)
= (Fh (u) − F̃h , ϕ) + (Fh∇ (u) − F̃h∇ , ∇ϕ)
≤ kFh (u) − F̃h kkϕk + kFh∇ (u) − F̃h∇ kk∇ϕk.
These estimates indicates that if Fh (u) and Fh∇ (u) are small, then the modeling error
u − uh is small and subgrid modeling is not needed. The quality of the corrected
solution ũh depends on how well we can approximate (Fh (u), Fh∇ (u)) by (F̃h , F̃h∇ ).
If the difference (Fh (u), Fh∇ (u)) − (F̃h , F̃h∇ ) is small, then the modeling error e − ẽh
is small.
For the dynamic subgrid modeling teqnique to work, it is vital to control the
numerical errors. First, if the numerical errors dominate, reduction of the modeling
errors will not significally reduce the total error. Secondly, the quality of the approximation FH (u) ≈ FH (uh ), and thus the whole extrapolation procedure, is affected
by the numerical errors in the computation of uh . Solving the problem (4.28) numerically by a Galerkin method corresponds to finding a function Uh ∈ Vh , where Vh is a
finite dimensional subspace of V , such that ah (Uh , v) = fh (u) for all v ∈ Vh . We will
in the following differentiate between the numerical discretization error arising in
solving the Galerkin equation, and the modeling error from not taking Fh (u) into account. We have that e = u−Uh = (u−uh)+(uh −Uh ) = eh +Eh , where eh represents
a modeling error and Eh a numerical error. Similarly, solving the corrected problem
(4.32) numerically corresponds to finding Ũh ∈ Vh such that ah (Ũh , v) = Fh (⊑)
for all v ∈ Vh , and we have ẽ = u − Ũh = (u − ũh ) + (ũh − Ũh ) = ẽh + Ẽh . If
keh k > kEh k, then subgrid modeling may be attempted with the goal of computing
a corrected solution Ũh , with kẽh k << keh k. We now present a posteriori error
estimates allowing us to estimate kek and kEh k, and thereby keh k, and similarly
kẽk and kẼh k, and hence kẽh k. We let w = Eh and w = Ẽh respectively, which
gives the error estimates
(Eh , Ψ) = ah (Eh , ϕ) = ah (uh , ϕ) − ah (Uh , ϕ) = fh (ϕ) − ah (Uh , ϕ)
= (Rh (Uh ), ϕ) + (Rh∇ (Uh ), ∇ϕ) ≤ kRh (Uh )kkϕk + kRh∇ (Uh )kk∇ϕk,
(Ẽh , Ψ) = ah (Ẽh , ϕ) = ah (ũh , ϕ) − ah (Ũh , ϕ) = F̃h (ϕ) − ah (Ũh , ϕ)
= (R̃h (Ũh ), ϕ) + (R̃h∇ (Ũh ), ∇ϕ) ≤ kR̃h (Ũh )kkϕk + kR̃h∇ (Ũh )kk∇ϕk,
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where Rh (w) = [f ]h − [β]h · ∇w − [α]h w, Rh∇ (w) = −[ǫ]h ∇w, R̃h (w) = [f ]h + F̃h −
[β]h · ∇w − [α]h w, and R̃h∇ (w) = −[ǫ]h ∇w + F̃h∇ .
Similarly, we can derive a posteriori error estimates for the total errors e = u−Uh
and ẽ = u − Ũh , in terms of computable residuals including both numerical and
modeling features. We let w = e and w = ẽ respectively, which gives the error
estimates
(e, Ψ) = ah (e, ϕ) = ah (u, ϕ) − ah (Uh , ϕ) = Fh (ϕ) − ah (Uh , ϕ)
= (R(Uh ), ϕ) + (R∇ (Uh ), ∇ϕ) ≤ kR(Uh )kkϕk + kR∇ (Uh )kk∇ϕk,
(ẽ, Ψ) = ah (ẽ, ϕ) = ah (u, ϕ) − ah (Ũh , ϕ) = Fh (ϕ) − ah (Ũh , ϕ)
= (R(Ũh ), ϕ) + (R∇ (Ũh ), ∇ϕ) ≤ kR(Ũh )kkϕk + kR∇ (Ũh )kk∇ϕk,
where R(w) = f − β · ∇w − αw, and R∇ (w) = −ǫ∇w.
Remark 4.1. Approximation of the dual solution, analytically or computationally,
is discussed in Eriksson et.al. 3 .
Remark 4.2. We can rewrite the residuals R(w) and R∇ (w) as
R(w) = Rh (w) + Fh (w) = R̃h (w) + Fh (w) − F̃h ,
R∇ (w) = Rh∇ (w) + Fh∇ (w) = R̃h∇ (w) + Fh∇ (w) − F̃h∇ .
Remark 4.3. The error measure of the form of an inner product of an error and
Ψ, the forcing function in the dual problem, includes a variety of possibilities. If we,
for example, let Ψ = χsi,k (where χsi,k denotes the characteristic function for the
subdomain si,k ) we get a local error representation for the local average of the error
at the midpoint of the subdomain si,k , and if we, for example, want to get a bound
on k[Ẽh ]h k, we choose Ψ = [Ẽh ]h and remember that [Ẽh ]h can be identified with
the L2 -projection of Ẽh onto Vi , which gives that (Ẽh , [Ẽh ]h ) = ([Ẽh ]h , [Ẽh ]h ) =
k[Ẽh ]h k2 . We may also estimate the error in linear functionals of u by using Riesz
representation theorem 9 .
5. Numerical Experiments
In this section we study the performance of the proposed subgrid model in some
numerical experiments. We use coefficients that are tensor products of the one
dimensional fractal functions of the form
X
ηγ,δ (x) = 1 +
γ 2−j(1/2+δ) ψj,l (x), x ∈ I = [0, 1],
(5.38)
j ≥ 0
l : x ∈ Ij,l
which are scale similar in the sense of the Corollary.
First we will use a very fine mesh (uniform quadratic with 512×512 elements) for
all computations, and we will assume the numerical errors to be neglible compared to
the modeling errors. We will denote by u the solution to the problem with the finest
possible resolved coefficients on this mesh, and we will use u as an approximation of
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1.25
1.2
1.15
1.1
1.05
1
0.95
0.9
0.85
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 1. ηγ,δ (x) for γ = 0.05 and δ = 0.5.
the exact solution to the problem in the computation of the error. To measure the
relative error in Uh and Ũh , projected onto the scale h, we define a “gain-factor”
GF , defined by
GF =
k[u − Uh ]h k
.
k[u − Ũh ]h k
(5.39)
We can also relate the reduction of the modeling error we get from using the extrapolated modeling residual to the reduction of the error we get by refining the
mesh, by introducing a “mesh-factor” M Fp , defined by
M Fp =
k[u − Uh/p ]h k
k[u − Ũh ]h k
.
(5.40)
For example, M F2 measures the relative improvement in Ũh compared to the improvement we get from refining the mesh uniformly once, that is, the improvement
we get by using four times as many nodes. In all the tests we are going to use the
cut-off scale h = 2−5 .
5.1. Elliptic Equations
As a model for an elliptic problem we consider Poisson’s equation
−∇ · (ǫ∇u) = f,
(5.41)
∂u
with the boundary conditions u|∂Ω− = ∂n
|∂Ω+ = 0, where ∂Ω− = {(x, y) : x =
0 or y = 0}, ∂Ω+ = {(x, y) : x = 1 or y = 1} and n is an unit outward normal from
Ω, and we formulate the corresponding variational problem: Find u ∈ V = {v ∈
H 1 (Ω) : v|∂Ω− = 0} such that
(ǫ∇u, ∇v) = (f, v),
∀v ∈ V.
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The solution to the simplified equation uh then satisfies
([ǫ]h ∇uh , ∇v) = ([f ]h , v),
∀v ∈ V,
and the exact solution u to (5.41) satisfies
([ǫ]h ∇u, ∇v) = ([f ]h , v) + (Fh∇ (u), ∇v),
∀v ∈ V,
where
Fh∇ (u) = [ǫ]h ∇u − ǫ∇u
is a modeling residual, and the projection of Fh∇ (u) onto Vi is defined by
F̄h∇ (u) = [ǫ]h [∇u]h − [ǫ∇u]h ,
which is of the form (3.18). Here we assume that ǫ has a simple fractal structure
and assuming that the solution u inherits this structure to some degree, we use the
Ansatz (3.25) on F̄h∇ (u): For given (x, y) ∈ Ω,
F̄h∇ (u)(x, y) ≈ C(x, y)hµ(x,y) ,
where C, µ : Ω → R2 are independent of the cut-off h. By approximating
∇
∇
F̄H
(u) ≈ F̄H
(uh ) for coarser scales H we can determine approximative C and
µ. The extrapolated approximation F̃h∇ of F̄h∇ (u) is then defined by
F̃h∇ (x, y) = C(x, y)hµ(x,y) ,
and we first define ũh to be the solution to the corrected problem
([ǫ]h ∇ũh , ∇v) = ([f ]h , v) + (F̃h∇ , ∇v),
∀v ∈ V.
We define the diffusion coefficient ǫ to be the tensor product of ηγ,δ : ǫ(x, y) =
ηγ,δ (x)ηγ,δ (y), with γ = δ = 0.1. For the computations we use a standard Finite
element method FEM with bilinear elements. When we compute uh and ũh for
f (x) = 1, and extrapolate from H = 2h and Ĥ = 4h, we find that GF = 1.1,
which is not very impressive. We have M F2 = 0.73, so the improvement we get by
uniformly refining the mesh once is here greater than the improvement we get by
correcting the equation with the extrapolated modeling residual.
From Theorem 2 we know that the modeling error in ũh depends on Fh (u) − F̃h .
But the error Fh (u)− F̃h consists of two components: Fh (u)− F̄h (u) and F̄h (u)− F̃h ,
where the first component is the error we get from projecting Fh (u) onto Vi and
the second is an extrapolation error. To compare the relative importance of the
two errors we can eliminate the extrapolation error by using u instead of uh in the
above experiment. We then get GF = 9.9 and M F2 = 7.8, so the extrapolation
error is large, and we conclude that if we can reduce the extrapolation error the
total modeling error will be significally reduced. By increasing H and Ĥ, we hope to
reduce the extrapolation error since we anticipate that the approximations F̄H (u) ≈
F̄H (uh ) and F̄Ĥ (u) ≈ F̄Ĥ (uh ) should be more accurate on coarser scales H and
Ĥ. We find that for (H, Ĥ) = (4h, 8h) we get GF = 1.5 and M F2 = 0.99, so
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2.5
2
1.5
1
0.5
0
0.2
0
0
0.4
0.2
0.6
0.4
0.6
0.8
0.8
1
1
Fig. 2. The fractal coefficient ǫ(x, y).
now the modeling error in ũh is the same as for one uniform mesh refinement. By
increasing H and Ĥ even more we find that the modeling error is further reduced.
We summarize these results in Table 1.
(H, Ĥ)
(2h,4h)
(2h,8h)
(4h,8h)
(4h,16h)
(8h,16h)
GF
1.1
1.2
1.5
1.8
1.8
M F2
0.73
0.77
0.99
1.2
1.2
Table 1. Extrapolation level dependence, modeling errors.
Now we introduce numerical errors into the problem by computing on a uniform
quadratic mesh with 32×32 elements, corresponding to the modeling scale h = 2−5 .
Using the same ǫ as above and (H, Ĥ) = (4h, 8h), we get GF = 1.7 and M F2 = 1.1.
We give the results for different (H, Ĥ) in Table 2. We thus find that the subgrid
model based on a corrective force works for Poisson’s equation with a fractal ǫ, but
to get a significant improvement in L2 -norm we have to extrapolate from coarser
scales than (H, Ĥ) = (2h, 4h). This we also noted for the one dimensional case 11 ,
and one might view this result as an indication that it is hard to model a corrected
diffusion operator by a correction on the force.
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(H, Ĥ)
(2h,4h)
(2h,8h)
(4h,8h)
(4h,16h)
(8h,16h)
GF
1.2
1.3
1.7
2.1
2.1
17
M F2
0.78
0.84
1.1
1.3
1.3
Table 2. Extrapolation level dependence, including numerical errors.
5.2. Hyperbolic Equations
Now we consider the problem (2.1) when ǫ << β, for which the problem has a
hyperbolic character. We use a Streamline Diffusion SD method 4 to solve the
problem
−ǫ∆u + β · ∇u + αu = f,
(5.50)
∂u
with boundary conditions u|∂Ω− = ∂n
|∂Ω+ = 0, where ∂Ω− = {x ∈ ∂Ω : β(x) ·
n(x) < 0}, ∂Ω+ = {x ∈ ∂Ω : β(x) · n(x) > 0} and n is an unit outward normal
from Ω. ǫ is constant such that ǫ << kβk, and we have the corresponding simplified
problem
−ǫ∆uh + [β]h · ∇uh + [α]h uh = [f ]h ,
with the same boundary conditions as for (5.50). The exact solution u to the problem
(5.50) satisfies
−ǫ∆u + [β]h · ∇u + [α]h u = [f ]h + Fh (u),
where the modeling residual Fh (u) projected onto Vi is given by
F̄h (u) = [β]h · [∇u]h − [β · ∇u]h + [α]h [u]h − [αu]h
∂u
∂u
∂u
∂u
= [β1 ]h [ ]h − [β1 ]h + [β2 ]h [ ]h − [β2 ]h + [α]h [u]h − [αu]h
∂x
∂x
∂y
∂y
= F̄h1 (u) + F̄h2 (u) + F̄h3 (u).
(5.53)
Now we let β = (β1 , β2 ), where β1 (x, y) = β2 (x, y) = ηγβ ,δβ (x)ηγβ ,δβ (y), and
α(x, y) = ηγα ,δα (x)ηγα ,δα (y). We set γβ = 0.2, δβ = 0.25, γα = 0.3, δα = 0.08,
and ǫ = 10−3 . Following the discussion in the previous sections, the simple fractal
structure of the coefficients α and β, together with the assumption that the solution inherits this structure to some degree, motivates an Ansatz on F̄h (u): For given
(x, y) ∈ Ω,
F̄hk (u)(x, y) ≈ Ck (x, y)hµk (x,y) ,
k = 1, 2, 3,
where Ck , µk : Ω → R are independent of the cut-off h. By computing approximak
k
tions F̄H
(u) ≈ F̄H
(uh ) on coarser scales H and Ĥ we can independently determine
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4h
8h
2h
4h
h
h
h/2
h/4
Fig. 3. Mesh hierarchy corresponding to scales 8h, 4h, ..., h/4.
(Ck , µk ) without using more scales than if we just had to determine one set of parameters (C, µ). We denote the extrapolated approximations F̃hk , and we denote the
sum F̃h . We define the corrected solution ũh to be the solution to the problem
−ǫ∆ũh + [β]h · ∇ũh + [α]h ũh = [f ]h + F̃h .
First we consider only modeling errors, making the numerical errors neglible compared to the modeling errors by computing on the fine mesh. We use the cut-off
√
h = 2−5 ≈ ǫ, and when extrapolating from (H, Ĥ) = (2h, 4h) we get GF = 1.9
and M F2 = 1.7. This means that the modeling error in the corrected solution ũh
is less than the modeling error in a non corrected solution on a uniformly refined
mesh, so ũh corresponds to an “extrapolation of uh beyond the unresolved scale
h/2”. We decompose Fh (u) − F̃h into (Fh (u) − F̄h (u)) + (F̄h (u) − F̃h ), where the
first component is the error we get from projecting Fh (u) onto Vi and the second
is an extrapolation error. By eliminating the extrapolation error, by using F̄h (u)
instead of F̄h (uh ), we find that also for this hyperbolic problem the extrapolation
error is significant. By comparing the error F̄h1 (u) − F̃h1 and F̄h2 (u) − F̃h2 to the error
F̄h3 (u) − F̃h3 , we find that the errors in the corrective forces that are based on ∇u are
a lot more significant than the error in the corrective force that is based on u. This
was also observed for the one dimensional problem 11 . In Figure 4 we have plotted
1
1
F̄Ĥ
(uh ), F̄H
(uh ) and F̃h1 , and in Figure 5 the extrapolated modeling residual F̃h is
plotted, where we have extrapolated from scales 2h and 4h. In Hoffman et.al. 11 ,
two ways to reduce the extrapolation error were suggested. One is to extrapolate
from coarser scales than H = 2h and Ĥ = 4h. In this case we assume that the loss
of accuracy we get from using coarser scales is dominated by the gain we get from
the fact that ∇uh better resembles ∇u on the coarser scales, and we further assume
that the scale regularity of the problem is similar on the scale Ĥ. By extrapolating
from H = 4h and Ĥ = 8h we get GF = 7.5 and M F4 = 2.3, so now ũh corresponds
to an “extrapolation of uh beyond the unresolved scale h/4”.
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0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1
−0.02
0
0.8
0.6
0.2
0.4
0.4
0.6
0.2
0.8
1
0
1 (u ) and F̃ 1 .
Fig. 4. F̄ 1 (uh ), F̄H
h
h
Ĥ
0.05
0.04
0.03
0.02
0.01
0
1
0.8
0.6
1
0.8
0.4
0.6
0.4
0.2
0.2
0
0
Fig. 5. The extrapolated modeling residual F̃h .
For the special cases when β = (β1 , 0) or β = (0, β2 ) we can avoid to base
the computation of the corrective force on ∇u at all. In the problem (5.50), with
β = (β1 , 0), we can eliminate ∇u from (5.53) by using (5.50) and neglecting the
terms including ǫ,
∂u
∂u h
] − [β1 ]h + [α]h [u]h − [αu]h
∂x
∂x
h 1
= [β1 ] [ (f − αu + ǫ∆u)]h − [f − αu + ǫ∆u]h + [α]h [u]h − [αu]h
β1
h
≈ [β1 ] [f /β1 ]h − [f ]h + [α]h [u]h − [β1 ]h [(αu)/β1 ]h .
F̄h (u) = [β1 ]h [
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But this method is, as we remarked, only applicable when β = (β1 , 0) or β = (0, β2 ),
and ǫ has to be small enough, so that the approximation of neglecting the terms
involving ǫ is justified.
We then introduce numerical errors by computing on a quadratic mesh with 32×
√
32 elements, corresponding to the cut-off scale h = 2−5 (h ≈ ǫ). By extrapolating
from H = 2h and Ĥ = 4h we get GF = 1.6 and M F2 = 0.91, so the modeling error
in the corrected solution approximately corresponds the modeling error in a non
corrected solution on the scale h/2. We find that by extrapolating from greater H
and Ĥ, up to a certain level, we further reduce the error in ũh (this is summarized
in Table 3).
error
modeling
modeling +
numerical
(H, Ĥ)
(2h,4h)
(2h,8h)
(4h,8h)
(4h,16h)
(8h,16h)
(2h,4h)
(2h,8h)
(4h,8h)
(4h,16h)
(8h,16h)
GF
2.1
2.4
7.5
7.2
1.1
1.6
1.8
3.9
5.5
1.4
M F2
1.2
1.4
4.5
4.3
0.63
0.91
1.0
2.2
3.1
0.79
M F4
0.65
0.75
2.3
2.2
0.33
0.47
0.52
1.1
1.6
0.40
Table 3. Extrapolation level dependence, with and without numerical errors.
Remark 5.1. In the initial study of this paper, the extrapolation is of a simple
form based on the Ansatz 3.25, and the approximation FH (u) ≈ FH (Uh ). In a
continuation of this work 7 , we refine the extrapolation procedure and present an
extrapolation formula that takes into consideration that the approximation FH (u) ≈
FH (Uh ) may be of a poor quality since Uh does not contain any scales finer than h,
whereas the exact solution u contain a range of scales finer than h.
6. Conclusions
In this paper we investigated the feasibility of extrapolating a modeling residual
to model the subgrid scales, in the case of a convection-diffusion-reaction problem
with features on a range of scales in two dimensions. We assumed some “scale
similarity” of the coefficients, and for this case we motivated an Ansatz on the
modeling residual using a Haar MRA. We then showed that it is possible to compute
the modeling residual on coarse scales with a fine scale computed solution without
subgrid model as a reference, and then extrapolate the modeling residual to the
computational scale. We presented error estimates for the corrected and the non
November 20, 2003 13:58 WSPC/INSTRUCTION FILE
hoffman
Subgrid modeling for convection-diffusion-reaction using a Haar MRA
21
corrected solutions, both for the numerical and the modeling errors. We showed
in numerical experiments for the hyperbolic model problem that by extrapolating
from 2h and 4h, where h is the computational scale, the sum of the numerical and
modeling errors in the corrected solution typically corresponded to a non corrected
solution on the scale h/2, and by extrapolating from 4h and 8h the corrected solution
corresponded to a non corrected solution on the scale h/4. For the elliptic problem,
extrapolating from 4h and 8h typically corresponded to a non corrected solution on
the scale h/2.
It was further noted that the extrapolation procedure was more efficient when
the corrective force was based on the computed solution, and not on the gradient
of the computed solution.
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