Title : Alternate Mixed Method: Old Wine in a New Bottle - IRCC

Alternate Mixed Method: Old Wine in a New 

Bottle 

Amiya Kumar Pani 1 

akp@math.iitb.ac.in 

Industrial Mathematics Group 

Department of Mathematics 

IIT Bombay, Powai 

Mumbai-400 076 

August 17, 2011 

1 Collaborators: G. Fairweather, R.K.Sinha, Arul Veda Manickam ,Kannan 

Moudgalya, M.A. Ali and A.K. Otta 

1 / 37 IRCC, IITB Alternate Mixed Method

Sketch of the Talk 

On Computational Mathematics 

A new way of looking at Mathematics 

Broad Focus 

Why Mixed Method 

An Example in Enhanced Oil Recovery 

To illustrate with simpler Problem 

Classical Mixed Method (p, u) 

Formulation: What we have gained & What we have lost 

LBB Consistency Condition 

Order of Convergence 

Disadvantages 


Sketch of the Talk (contd.) 

Proposed Mixed Method 

Basic Idea: symmetrization 

Old wine in a new bottle : justification 

Order of convergence 

Comparision 

Modified Method 

Non-conformity 

Improved rate of convergence 

Summary and Extension 



☞ When a given problem is approximated using a finite 

machine ( machine with a finite precession), we commit 

round off errors or truncation errors at every stage of 

computation. 

☞ It is, therefore, essential to have a control on these errors. 

☞ To illustrate the effect of round off or truncation error, 

consider solving 

AX = b, (1) 

where b is a given vector and A is a Hilbert matrix, which 

is given by 

1 

A ij = 

i + j − 1 . 



☞ When a given problem is approximated using a finite 

machine ( machine with a finite precession), we commit 

round off errors or truncation errors at every stage of 

computation. 

☞ It is, therefore, essential to have a control on these errors. 

☞ To illustrate the effect of round off or truncation error, 

consider solving 

AX = b, (1) 

where b is a given vector and A is a Hilbert matrix, which 

is given by 

1 

A ij = 

i + j − 1 . 


On Computational Mathematics (contd.) 

In other words 

⎡ 

A n×n = 

⎢ 

⎣ 

1 

1 

3 

· · · 

n 

1 

1 

4 

· · · 

n+1 

1 

1 

5 

· · · 

n+2 

· · · · · · · · · · · · · · · 

1 1 1 

1 

n n+1 n+2 

· · · 

2n−1 

1 

1 

2 

1 1 

2 3 

1 1 

3 4 

⎤ 

⎥ 

⎦ 



☞ It is easy check that 

det A ≠ 0, 

and hence, the matrix A is nonsingular. 

☞ Therefore, (1) yields 

X := A −1 b. (2) 

☞ When you compute with finite machine: a small change in 

b gives rise to a drastic change in the solution. 

☞ For example, consider 

b ′ = (1, 0, 0, · · · , 0) 1×50 

and a slight perturbation of b: 

˜b ′ = (1 + 10 −3 , 0, 0, · · · , 0) 1×50 . 



☞ It is easy check that 

det A ≠ 0, 

and hence, the matrix A is nonsingular. 

☞ Therefore, (1) yields 

X := A −1 b. (2) 

☞ When you compute with finite machine: a small change in 

b gives rise to a drastic change in the solution. 

☞ For example, consider 

b ′ = (1, 0, 0, · · · , 0) 1×50 

and a slight perturbation of b: 

˜b ′ = (1 + 10 −3 , 0, 0, · · · , 0) 1×50 . 



☞ Now, 

‖b − ˜b‖ max := max 

1≤i≤50 |b i − ˜b i | ≤ 10 −3 . 

For computing solution of 

AX = b, and A˜X = ˜b, 

using a finite machine, we observe that 

‖X − ˜X‖ max = max 

1≤i≤50 |X i − ˜X i | ≤ 2.9329 × 10 6 . 

small chage in b gives rise to drastic change in the 

solution. 

☞ Therefore, if we use low end machine and then high end 

machine, there will be be a drastic change in the solution 

and it is not acceptable as an approximate solution. 



☞ Now, 

‖b − ˜b‖ max := max 

1≤i≤50 |b i − ˜b i | ≤ 10 −3 . 

For computing solution of 

AX = b, and A˜X = ˜b, 

using a finite machine, we observe that 

‖X − ˜X‖ max = max 

1≤i≤50 |X i − ˜X i | ≤ 2.9329 × 10 6 . 

small chage in b gives rise to drastic change in the 

solution. 

☞ Therefore, if we use low end machine and then high end 

machine, there will be be a drastic change in the solution 

and it is not acceptable as an approximate solution. 



☞ To probe a bit: it is the effect of round off error. 

Therefore, a new way of looking at Mathematics 

has emerged in the last fifty or sixty 

years. A part from Pure and Applied Mathematics, 

now a new branch called ‘Computational 

Mathematics’ has developed. 

☞ In this talk, we concentrate on ’Computational PDEs’. 


Computational PDEs 

Broad Focus: 

To provide mathematical justifications in terms of 

stability and convergence to widely used numerical 

schemes for computational PDEs. 

To design and develop reliable and efficient algorithms 

for numerical solutions to PDEs. 

By reliability, we mean that for a given tolerance and 

measurement, the computed solution stays near to the 

exact unknown solution that too within the prescribed 

tolerance with respect to the given measurement. 

By efficient, we understand that this can be achieved with 

minimal computational effort. 

Theme of this talk: propose a new method with proper 

mathematical justifications. 


Computational PDEs 

Broad Focus: 

To provide mathematical justifications in terms of 

stability and convergence to widely used numerical 

schemes for computational PDEs. 

To design and develop reliable and efficient algorithms 

for numerical solutions to PDEs. 

By reliability, we mean that for a given tolerance and 

measurement, the computed solution stays near to the 

exact unknown solution that too within the prescribed 

tolerance with respect to the given measurement. 

By efficient, we understand that this can be achieved with 

minimal computational effort. 

Theme of this talk: propose a new method with proper 

mathematical justifications. 


To motivate through an example from Enhanced Oil 

Recovery 

In tertiary oil recovery an inexpensible fluid is injected into the 

reservoir to mix with the hydrocarbon (Oil) and then push the 

mixture towards the production wells. 


Enhanced Oil Recovery (contd.) 

Assuming that there is no volume change due to mixing and the 

fluid is slightly compressible , the mathematical model gives 

rise to the following system of PDEs: 

For a given T > 0, find a pair (p,c) such that 

✓ 

d(c) ∂p 

∂t 

− ∇.(a(c)∇p) = q, (x, t) ∈ Ω × J 

φc t + b(c) ∂p 

✒ 

∂t 

+ u.∇c − ∇.(D(u)∇c) = g(c) 

with no flow boundary conditions and initial condition 

✏ 

✑ 

p(x, 0) = p 0 (x), c(x, 0) = c 0 (x), x ∈ I. 


Enhanced Oil Recovery (contd.) 

• Ω is a bounded domain in IR 2 , J = (0, T ). 

• u is the Darcy velocity, 

u = −(a(c)∇p). 

• a(c) = k(x) 

µ(c) 

, k(x)- the permeability, µ(c) - viscosity. 

• φ -the porosity 

• q is the total external flow. 

• D = D(x, u) is diffusion coefficient. 

• ˜c is a function specified at injection point and equals to c at 

production point, c = c 1 = 1 − c 2 , where c 1 and c 2 is the 

concentrations of the two fluid components, 0 ≤ c ≤ 1. 

• g(c) = (c − ˜c)q, 

• c 0 (x) is a function must be specified. 


Quantity of Interest 

Quantity of Interest: Concentration c 

(as more accurate evaluation of c is related 

to ‘How much oil (hydrocarbon) can 

be extracted from the Reservoir’) 

☞ For accurate evaluation of concentration c, we need more 

accurate evaluation of u as u not p appears in the 

concentration equation. 

Rule of the game: consider both p and u as primary 

variables,that is, comute both simultaneously. 


Quantity of Interest 

Quantity of Interest: Concentration c 

(as more accurate evaluation of c is related 

to ‘How much oil (hydrocarbon) can 

be extracted from the Reservoir’) 

☞ For accurate evaluation of concentration c, we need more 

accurate evaluation of u as u not p appears in the 

concentration equation. 

Rule of the game: consider both p and u as primary 

variables,that is, comute both simultaneously. 


To illustrate with Simpler Problem 

Find p on Ω × (0, T ] such that p satisfies 

✗ 

✔ 

p t − ∇ · (a(x)∇p) = f (x, t), (x, t) ∈ Ω × J, (3) 

✖ 

p = 0, (x, t) ∈ ∂Ω × J, 

p(x, 0) = p 0 , x ∈ Ω. 

Ω be a bounded domain in R d , d = 2, 3 with smooth 

boundary ∂Ω. 

a ≥ a 0 > 0 

p t = ∂p 

∂t 

and J = (0, T ] with T < ∞. 

✕ 


Why Mixed Methods ? 

☞ To approximate p (temperature distribution) and its flux 

(heat flux) u = −a ▽ p through standard numerical 

procedures (FDM or FEM): 

Compute an approximation to p that is say p h . 

Post-process it to obtain an approximation to u that is 

u h = a ▽ p h . 

☞ Outcome: Inaccurate approximation to u. 

(due to numerical differentiation or/and multiplication by 

rough coefficient a). 


Why Mixed Methods ? 

☞ To approximate p (temperature distribution) and its flux 

(heat flux) u = −a ▽ p through standard numerical 

procedures (FDM or FEM): 

Compute an approximation to p that is say p h . 

Post-process it to obtain an approximation to u that is 

u h = a ▽ p h . 

☞ Outcome: Inaccurate approximation to u. 

(due to numerical differentiation or/and multiplication by 

rough coefficient a). 


Objective of Mixed Methods. 

☞ Reformulate the problem with both p and u as primary 

variables. 

☞ Split (3) as a system ✓ 

✒ 

with p(x, 0) = p 0 . 

αu = ∇p, α = 1 a , 

∂p 

∂t 

− ∇ · u = f . 

✏ 

✑ 

☞ Weak Formulation. Let V = L 2 (Ω) : space of squre 

integrable functions, and 

{ 

} 

W = H(div, Ω) = w : w ∈ (L 2 (Ω)) d , ∇ · w ∈ L 2 (Ω) 

with norm 

‖w‖ H(div,Ω) = (‖w‖ 2 + ‖∇ · w‖ 2 ) 1/2 . 


Weak Formulation 

Find (p, u) : (0, T ] ↦→ V × W such that 

(αu, w) + (∇ · w, p) = 0, w ∈ W, 

( ∂p 

∂t 

, v) − (∇ · u, v) = (f , v), v ∈ V . (2b) 

(2a) 

Galerkin Procedure. Choose the finite element spaces V h of 

V and W h of W . Seek a pair (p h , u h ) : (0, T ] ↦→ V h × W h such 

that 

(αu h , w h ) + (∇ · w h , p h ) = 0, w h ∈ W h , 

( ∂p h 

∂t 

, v h ) − (∇ · u h , v h ) = (f , v h ), v h ∈ V h . 


Error Analysis. 

☞ LBB Consistency Condition: to compensate nonsymmetric 

(v h ,∇·w h ) 

sup 

‖w h ‖ 

≥ β‖v 

H(div;Ω) 

h ‖ V , v ∈ V h . 

w h ∈ W h /{0} 

☞ Restriction on the construction of finite dimensional 

subspaces V h and W h . 

∇ · W h = V h 


Raviart-Thomas-Nedelec Spaces. 

T h : quasi-uniform triangulation of Ω (assuming Ω is a polygon) 

by triangles K of diameter at most h. 


Raviart-Thomas-Nedelec Spaces. 

☞ Now V h = {v h ∈ V : v h | K ∈ P r−1 (K ), K ∈ T h }. 

☞ P j - a polynomial of degree at most j. 

☞ Approximation property: inf vh ∈V h 

‖v − v h ‖ ≤ Ch r ‖v‖ r . 

☞ For W h , set Q(K ) = [P r−1 ] 2 ⊕ Span(xP r−1 (K )) 

(space of incomplete polynomials lying in between P r and 

P r−1 ) and define 

W h = {w h ∈ H(div, Ω) : w h | K ∈ Q(K ), K ∈ T h }. 

☞ Approximation property: inf wh ∈W h 

‖w − w h ‖ ≤ Ch r ‖w‖ r . 

☞ This construction preserves the relation 

∇ · W h = V h . 


Convergence. 

For example: when r = 1 

V h = {v ∈ L 2 : v| K ∈ P 0 (K ), K ∈ T h }. 

and 

W h = {w ∈ H(div; Ω) : w| K = (a 1,K +b K x 1 , a 2,K +b K x 2 ) K ∈ T h }. 

(Johnson & Thomeé , RAIRO 1981) 

‖(p h − p)(t)‖ ≤ Ch r {‖p(t)‖ r + ∫ t 

0 ‖p t(τ)‖ r dτ} 

‖(u h − u)(t)‖ ≤ Ch r {‖p(t)‖ r+1 + ∫ t 

0 ‖p t(τ)‖ r dτ} 

For r = 1: Order of Convergence is One 


Disadvantages. 

❶ Due to quasi-uniformity condition, there is difficulty in 

applying adaptive methods. 

❷ LBB consistency condition must be satisfied, and this 

imposes severe restriction on the construction of finite 

element spaces 

❸ Computationally more simple and attractive C 0 -piecewise 

linear spaces to approximate both of p and u can not be 

used. 


Alternate Method. 

☞ Attempts in the Past. (Specially for Elliptic Problems) 

Use of some short of stabilizers (computationally more 

complex and time consuming). In fact, LBB-condition can 

be thought of as a stabilizer. 

☞ Main Culprit. Non-symmetric form of 

(∇ · w h , p h ), (∇ · u h , v h ). 

☞ Possible Remedy: Try to symmetrize. 

To provide an answer to the 14th open problem cited in a 

workshop on Free and Moving Boundary Problems held in 

Berlin (1977), a kind of symmetrization technique is used 

by us for one phase unidimensional Stefan problem (Pani 

and Das, IMA J. Numer. Anal. 1989, 1991,...) 

Spurt of Activities around 1993 for elliptic problems using 

Least Square Method. 


Proposed Scheme 

☞ Split (3) as two first order system 

☞ Try to symmetrize. 

u = a∇p, 

p t − ∇·u = f . 

(4a) 

(4b) 

☞ Mulptiply (4a) by ∇v and (4b) by ∇ · w. Then integrate over 

Ω. Write (p t , ∇ · w) = −(∇p t , w) = −(αu t , w), where 

α = 1/a. 

☞ Weak Formulation. Find a pair {p, u} : J ↦→ H 1 0 × W 

satisfying for w ∈ W 

(a∇p, ∇v) = (u, ∇v) , v ∈ H 1 0 , (5a) 

(αu t , w) + (∇·u, ∇·w) = − (f , ∇·w) .(5b) 


Galerkin Approximation. 

☞ For all K ∈ T h 

V h = { v h ∈ C 0 (Ω) : v h | K ∈ P k (K ), v h = 0 on ∂Ω } 

W h = {q h ∈ W : (q h ) i | K ∈ P r (K ), i = 1, · · · , d} 

☞ Approximation property: 

inf vh ∈V h 

‖v − v h ‖ + h‖∇(v − v h )‖ ≤ Ch k+1 ‖v‖ k+1 . 

inf wh ∈W h 

‖w − w h ‖ + h‖∇(w − w h )‖ ≤ Ch r+1 ‖v‖ r+1 . 

☞ Determine a pair {p h , u h } : J ↦→ V h × W h satisfying for 

w h ∈ W h 

(a∇p h , ∇v h ) = (u h , ∇v h ) , v h ∈ V h , (6a) 

(αu ht , w h ) + (∇·u h , ∇·w h ) = − (f , ∇·w h ) (6b) 


Semidiscrete Error Estimates. 

with appropriately chosen initial pair {p h (0), u h (0)} to be 

defined later. 

Theorem 

With u 0 = a∇p 0 , assume that 

‖u 0 − u 0h ‖ H(div,Ω) ≤ Ch r ‖u 0 ‖ r+1 . 

Then there is a constant C independent of h such that 

‖∇(p − p h )(t)‖ + ‖∇ · (u − u h )(t)‖ ≤ C(u, p) h min(r,k) , 

‖(p − p h )(t)‖ + ‖(u − u h )(t)‖ ≤ C(u, p) h min(r,k+1) . 


Remark. 

✍ For k = r, the error estimate ‖∇ · (u − u h )(t)‖ is optimal. 

However, the rate of convergence in L 2 -norm is not 

optimal. This is due to the presence of ∇ · ρ term in the 

right hand side of (8b). It is possible to choose the auxiliary 

projection ũ I through 

(∇ · (u − ũ I ), ∇ · w h ) = 0, w h ∈ W h , 

so that the term containing ∇ · ρ is zero. But it is not 

possible to obtain optimal L 2 -estimate for ρ t term. 


☞ Note that Aubin-Nitsche duality argument requires the 

following regularity result 

‖Φ‖ 2 ≤ C‖g‖ 

(∗) 

for the associated operator equation 

☞ This may not be true. But for 

∇(∇ · Φ) = g. 

−∆Φ = ∇ × ∇ × Φ − ∇(∇ · Φ) 

the regularity condition (*) holds. 

Question: 

‘Is it possible to modify the formulation 

so that an additional term ∇ × ∇ × Φ 

is introduced?’ 


☞ Note that Aubin-Nitsche duality argument requires the 

following regularity result 

‖Φ‖ 2 ≤ C‖g‖ 

(∗) 

for the associated operator equation 

☞ This may not be true. But for 

∇(∇ · Φ) = g. 

−∆Φ = ∇ × ∇ × Φ − ∇(∇ · Φ) 

the regularity condition (*) holds. 

Question: 

‘Is it possible to modify the formulation 

so that an additional term ∇ × ∇ × Φ 

is introduced?’ 


Modified Method 

☞ Observe that ∇ × (∇φ) = 0. 

☞ Rewrite (1) as 

p t − ∇·u = f , (x, t) ∈ Ω × J, 

∇ × (αu) = 0, (x, t) ∈ Ω × J, (αu = ∇p), 

(n ∧ αu) = 0, (x, t) ∈ ∂Ω × J, p(0) = p 0 , 

n - the outward normal 

∧ - exterior product. 

☞ Weak Formulation: Let 

W = {w ∈ (H 1 ) d : n ∧ w = 0 on ∂Ω, d = 2, 3}. 

Seek a pair {p, u} : J −→ H0 1 × W such that 

(∇p, ∇v) = (αu, ∇v), v ∈ H0 1, 

(8a) 

(αu t , w) + A(u, w) = −(f , ∇ · w), w ∈ (H 1 ) d , (8b) 

where 

A(φ, w) = (∇ · φ, ∇ · w) + (∇ × αφ, ∇ × αw). 


Galerkin Formulation. 

☞ A(·, ·) satisfies the coercivity condition 

A(φ, φ ≥ µ 0 ‖φ‖ 2 1 , 

for some positive constant µ 0 , [refer to Naittaanmäki and 

Saranen (1981)]. 

☞ Galerkin Formulation. V h is defined earlier. 

W h = {w h ∈ C(¯Ω) d : (w h ) i | K ∈ P r (K ), i = 1, · · · , d, 

(n ∧ αw h ) = 0, at the nodes on ∂Ω}. 

Since (n ∧ αw h ) = 0 only at the boundary nodes, 

W h ⊂ W,(non-conforming method). 

☞ Determine a pair {p h , u h } ∈ V h × W h such that w h ∈ W h , 

(∇p h , ∇v h ) = (αu h , ∇v h ), v ∈ V h , 

(9a) 

(αu ht , w h ) + A(u h , w h ) = −(f , ∇ · w h ), (9b) 


Error Analysis 

Theorem (Theorem 2) 

Assume that u h (0) = ũ h (0) with u(0) = a∇p 0 . Then, there 

exists a positive constant C independent of h such that 

‖(p − p h )(t)‖ + ‖(u − u h )(t)‖ + h‖(p − p)(t)‖ 1 ≤ 

‖(u − u h )(t)‖ 1 

] 

≤ 

Ch min(k+1,r+1) 

Ch 

r 


Remarks. 

1 With k = r we have 

‖p − p h ‖ L ∞ (L 2 ) + ‖u − u h ‖ L ∞ ((L 2 ) d ) = O(hr+1 ). 

2 Compared to Johnson and Thomeé [1981] the present 

analysis yields better results for L 2 -estimate of u − u h with 

higher regularity assumption on the exact solution. Further, 

‖u − u h ‖ = O(h r+1 ) even if k ≠ r with k < r. 

3 When d = 2 that is Ω ⊂ R 2 and u h (0) = ũ h (0), then using 

Sobolev imbedding theorem and superconvergence 

estimates for the ∇ζ and ∇ξ, we have the following 

maximum norm estimates 

‖(p − p h )(t)‖ L ∞ ≤ C(log 1 h )hmin(k+1,r+1) 

provided we assume quasi-uniform condition on the finite 

element mesh. 


Summary & Concluding Remarks 

✌ The Galerkin approximation has the same rate of 

convergence as in the case of classical mixed finite 

element method, but without LBB consistency condition 

and also without quasiuniform assumption on the finite 

element mesh. 

✌ Since the estimate for ‖u − u h ‖ is not optimal in two- and 

three space dimensions, a modification is proposed and 

analyzed to derive sharper estimate in L 2 -norm. 


Contd. 

✌ Compared to classical classical mixed method, the present 

method allows to use two different finite element spaces for 

approximating p and its flux u. In particular, use of 

piecewise linear polynomial spaces yields O(h 2 ) order of 

convergence in both p − p h and u − u h in L 2 − norm. 

Although higher regularity on the solution is assumed, but 

better convergence results are proved specially for the flux. 


Contd. 

Our major results: 

✍ Amiya K. Pani (1998),Parabolic equation with non-self 

adjoint elliptic part, SIAM J. Numer. Anal. 35, pp.712–727. 

✍ Amiya K. Pani and Gareme Fairweather (2002), 

H 1 -Galerkin mixed finite element methods for parabolic 

integro-differential equations, IMA J. Numer. Anal. 22, pp. 

231–252. 

✍ Amiya K. Pani and Graeme Fairweather (2002), An 

H 1 -Galerkin mixed finite element method for an evolution 

equation with a positive type memory term, SIAM J. 

Numer. Anal. 40, pp. 1475–1490. 


Contd. 

✎ Rajen K. Sinha, Ajay K. Otta and Amiya K. Pani (2004), An 

H 1 -Galerkin mixed method for second order hyperbolic 

equations, International J. Numer. Anal and Modeling, 1, 

pp. 111-130. 

✎ S. Arul Veda Manickam, Kannan M. Moudgalya and A. K. 

Pani (2004), Higher order fully discrete scheme combined 

with H 1 -Galerkin mixed finite element method for 

semilinear reaction-diffusion equations, J. Applied 

Mathematics and Computing, 15, pp. 1-28. 

✎ M. A. Mohamed Ali, Ph.D. dissertation on ‘Numerical 

Methods for Enhanced Oil Recovery in Reservoir Studies’, 

August ’97. 


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