Download FDM_Error_analysis

2D Finite Difference
 Δu  f
u
in
Ω
 0 on Ω
1
h2
-1
-1
4

  hU i , j  f ij
-1
U 0 j  U i 0  U M , j  U i,M  0
5 point-scheme
5 point stencil
  hUi , j  11Ui , j   2 2Ui , j

1
h2
U
i 1, j
 U i 1, j  4U i , j  U i , j 1  U i , j 1
-1

Taylor Expansions in 2d
THEOREM—Taylor’s Theorem If u and its first n derivatives are continuous on
the closed interval between a and a+h, and is differentiable on the open
interval between a and a+h, then there exists a number c between a and a+h
such that
u(a  h)  u(a)  u' (a)h 
u ''( a )
2!
h  
2
u( n) (a)
n!
h 
n
u ( n1) ( c )
( n 1)!
Taylor’s 2D
u ( xi  h, y j )  u ( xi , y j )  u x ( xi , y j )h 
u xx ( xi , y j )
u ( xi , y j  h)  u ( xi , y j )  u y ( xi , y j )h 
u yy ( xi , y j )
2!
2!
h2 
u xxx ( ci , y j )
h2 
u yyy ( xi ,c j )
3!
3!
u ( xi  h1 , y j  h2 )  u ( xi , y j )  u x ( xi , y j )h1  u x ( xi , y j )h2 

 Oh
 21! u xx ( xi , y j )h12  2u xy ( xi , y j )h1 h2  u yy ( xi , y j )h22
3
1
 h23


h3
h3
hn1
2D Finite Difference
Lapidus, Leon, and George F.
Pinder. Numerical solution of partial
differential equations in science and
engineering. John Wiley & Sons,
2011.
Maximum Principle (Continuous Problem)
(DMP)
(PT)
Discrete Maximum Principle
Positive type scheme
If   hU ij  0, for
Eij , Wij , N ij , Sij  0
Pij  0
Pij  Eij  Wij  N ij  Sij
( xi , y j )  
max U  max U


 N ij
 Wij
Pij
 S ij
 Eij
(DD)
Diagonally Dominant
the sum of the absolute values of the
off-diagonal elements in one row is
bounded by the diagonal element in
that row
aii   aij
j i
Example
B
 I




I




I



 I

B
Chapter 4: Finite Difference Methods for Elliptic Equations
We consider
Au    (ku )  b  u  cu
   (ku )  b  u  cu  f in 
u  g in 
Example
Maximum Principle
 ( x, y )  x 2  y 2
   4  0
Consider the differential operator A , and assume
that u C 2 ( )
and c = 0
Au  0
If
Remarks:
its maximum is
located at (1,1)
max u  max u
in Ω


1) the maximum of u is attained at the boundary
2) if u has a maximum at an interior point of Ω, then u is constant
3) min principle is reduced to the max principle by looking at −u.
Stability Estimate (w.r.t the max-norm)
If
u C 2 (  )
there is a constant C such that
u
Remarks:
C()
 u
C ()
 C u
C()
The constant C depends on the coefficients of A but not on u.
RHS of the above inequality contains only the data of the problem
Chapter 4: Finite Difference Methods for Elliptic Equations
We consider
  hUi , j  11Ui , j   2 2Ui , j
Example
w( x, y)  x  x 2  y  y 2
Wij  w( xi , y j )
its maximum is
located at (1,1)
Discrete Maximum Principle
If U is such that
U attains its max
  hU i , j  0
for
for some
( xi , y j )  
Remark:
( xi , y j )  
Proof: assmume max at
interior  max at all neibhor
1) the maximum principle implies a stability estimate
Stability Estimate
For any U
mesh-function
Remarks:
there is a constant C such that
U   U   C  hU 
The constant C independent of U and h. but not on u.
Proof:
Truncation Errors
Remarks:
for u  C 4
 hu ( xi , y j )  u ( xi , y j )  Ch 2 u C 4
Truncation Error
This measures how well the exact solution of the
differential equation (which we do not know)
satisfies the difference equations
ΔhU ij  f ij  0
 ij  Δhu ( xi , y j )  f ( xi , y j )  0
 Δhu ( xi , y j )  Δu ( xi , y j )
 Δu  f

U i1, j U i1, j  4U i , j U i , j 1 U i , j 1
h2
 f ij
Error Estimate
2
1
 Δu  f in Ω
  hU i , j  f ij
u  0 on Ω
U 0 j U i 0 U M , j U i ,M  0
define:
eij  U ij  u ( xi , y j )
  h eij   ij
error
eij    h   ij
1
Theorem 4.2 (Error Estimate)
Let
Let
Then
u
U
be the solution of
be the solution of
1
2
U  u   Ch2 u C 4
Proof: Apply stability result on
eij  KCh 2
truncation
stability
stability
eij
Then use truncation error
U   U   C  hU 
eij  A1 Ch 2
A 1  C
Independent of h
as h 0
Theorem
stability plus consistency
implies convergence
Maximum Principle (Continuous Problem)
Lh
(PT)
satisfies (DMP)
Lh of positive type
Discrete Maximum Principle
Eij , Wij , N ij , Sij  0
Pij  0
LhU ij  LhVij , in h 
U ij  Vij
If 

 U ij  Vij , in h  everywhere
Pij  Eij  Wij  N ij  Sij
 N ij
 Wij
Pij
 S ij
 Eij
(DD)
Diagonally Dominant
the sum of the absolute values of the
off-diagonal elements in one row is
bounded by the diagonal element in
that row
aii   aij
j i
Example
B
 I




I




I



 I

B
curved boundary
curved boundary