finite element convergence analysis for non-smooth solutions
TRANSCRIPT
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Finite Element Convergence Analysis fornon-smooth solutions
Abiy Dejenee ZelekeBahru Tsegaye Leyew
Supervisor: Walter Zulehner
Johannes Kepler University Linz, Austria
7th December 2010
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Outline
Finite Element Method- Simple 1D Example
Discretization ErrorInterpolation theoremProblemK-functional
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions2 / 17
![Page 3: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/3.jpg)
Outline
Finite Element Method- Simple 1D Example
Discretization ErrorInterpolation theoremProblemK-functional
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions2 / 17
![Page 4: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/4.jpg)
Outline
Finite Element Method- Simple 1D Example
Discretization ErrorInterpolation theoremProblemK-functional
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions2 / 17
![Page 5: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/5.jpg)
Outline
Finite Element Method- Simple 1D Example
Discretization ErrorInterpolation theoremProblemK-functional
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions2 / 17
![Page 6: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/6.jpg)
Outline
Finite Element Method- Simple 1D Example
Discretization ErrorInterpolation theoremProblemK-functional
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions2 / 17
![Page 7: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/7.jpg)
Simple 1D ODE
Consider the following BVP−u
′′(x) = fα(x), x ∈ (0,1)
u(0) = u(1) = 0,
where fα(x) = xα, fα ∈ L2(0,1) for α > −1/2.Exact solution:
u(x) =x − xα+2
α2 + 3α + 2.
Let
V = H10 (0,1) = υ ∈ H1(0,1) : υ(0) = υ(1) = 0
u ∈ V ,∫ 1
0u′υ′dx =
∫ 1
0fυdx , ∀υ ∈ V
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions3 / 17
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Simple 1D ODE
Consider the following BVP−u
′′(x) = fα(x), x ∈ (0,1)
u(0) = u(1) = 0,
where fα(x) = xα, fα ∈ L2(0,1) for α > −1/2.Exact solution:
u(x) =x − xα+2
α2 + 3α + 2.
Let
V = H10 (0,1) = υ ∈ H1(0,1) : υ(0) = υ(1) = 0
u ∈ V ,∫ 1
0u′υ′dx =
∫ 1
0fυdx , ∀υ ∈ V
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions3 / 17
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Simple 1D ODE
Consider the following BVP−u
′′(x) = fα(x), x ∈ (0,1)
u(0) = u(1) = 0,
where fα(x) = xα, fα ∈ L2(0,1) for α > −1/2.Exact solution:
u(x) =x − xα+2
α2 + 3α + 2.
Let
V = H10 (0,1) = υ ∈ H1(0,1) : υ(0) = υ(1) = 0
u ∈ V ,∫ 1
0u′υ′dx =
∫ 1
0fυdx , ∀υ ∈ V
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions3 / 17
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Finite Element Method
The weak formulation is
u ∈ V , a(u, υ) = `(υ) ∀υ ∈ V (1)
Existence of a unique solution is guaranteed by theLax-Milgram LemmaPartition the domain I = [0,1] into N parts as0 = x0 < x1 < ... < xN = 1
- The points xi ,0 ≤ i ≤ N are called nodes,- The subintervals Ii = [xi−1, xi ],1 ≤ i ≤ N are called
elementsDenote hi = xi − xi−1 and the mesh parameterh = max1≤i≤N hi
Approximate solution will be sought in the spaceVh = υh ∈ V : υh|Ii
∈ P1(Ii),1 ≤ i ≤ N: note that, giventhe properties of the Sobolev space H1(I), we also haveυh ∈ C (I)
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions4 / 17
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Finite Element Method
The weak formulation is
u ∈ V , a(u, υ) = `(υ) ∀υ ∈ V (1)
Existence of a unique solution is guaranteed by theLax-Milgram LemmaPartition the domain I = [0,1] into N parts as0 = x0 < x1 < ... < xN = 1
- The points xi ,0 ≤ i ≤ N are called nodes,- The subintervals Ii = [xi−1, xi ],1 ≤ i ≤ N are called
elementsDenote hi = xi − xi−1 and the mesh parameterh = max1≤i≤N hi
Approximate solution will be sought in the spaceVh = υh ∈ V : υh|Ii
∈ P1(Ii),1 ≤ i ≤ N: note that, giventhe properties of the Sobolev space H1(I), we also haveυh ∈ C (I)
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions4 / 17
![Page 12: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/12.jpg)
Finite Element Method
The weak formulation is
u ∈ V , a(u, υ) = `(υ) ∀υ ∈ V (1)
Existence of a unique solution is guaranteed by theLax-Milgram LemmaPartition the domain I = [0,1] into N parts as0 = x0 < x1 < ... < xN = 1
- The points xi ,0 ≤ i ≤ N are called nodes,- The subintervals Ii = [xi−1, xi ],1 ≤ i ≤ N are called
elementsDenote hi = xi − xi−1 and the mesh parameterh = max1≤i≤N hi
Approximate solution will be sought in the spaceVh = υh ∈ V : υh|Ii
∈ P1(Ii),1 ≤ i ≤ N: note that, giventhe properties of the Sobolev space H1(I), we also haveυh ∈ C (I)
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions4 / 17
![Page 13: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/13.jpg)
Finite Element Method
The weak formulation is
u ∈ V , a(u, υ) = `(υ) ∀υ ∈ V (1)
Existence of a unique solution is guaranteed by theLax-Milgram LemmaPartition the domain I = [0,1] into N parts as0 = x0 < x1 < ... < xN = 1
- The points xi ,0 ≤ i ≤ N are called nodes,- The subintervals Ii = [xi−1, xi ],1 ≤ i ≤ N are called
elementsDenote hi = xi − xi−1 and the mesh parameterh = max1≤i≤N hi
Approximate solution will be sought in the spaceVh = υh ∈ V : υh|Ii
∈ P1(Ii),1 ≤ i ≤ N: note that, giventhe properties of the Sobolev space H1(I), we also haveυh ∈ C (I)
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions4 / 17
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FEM Continued
For the basis functions we choosefor i = 1, . . . ,N − 1
φi(x) =
x−xi−1
hi, xi−1 ≤ x ≤ xi ;
xi+1−xhi+1
, xi ≤ x ≤ xi+1, ;0, otherwise.
The basis functions Vh = spanφi ,1 ≤ i ≤ N are linearlyindependent.
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions5 / 17
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FEM Continued
For the basis functions we choosefor i = 1, . . . ,N − 1
φi(x) =
x−xi−1
hi, xi−1 ≤ x ≤ xi ;
xi+1−xhi+1
, xi ≤ x ≤ xi+1, ;0, otherwise.
The basis functions Vh = spanφi ,1 ≤ i ≤ N are linearlyindependent.
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions5 / 17
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FEM Continued
The FEM thus becomes:
uh ∈ Vh,
∫ 1
0(u′
hυ′
h)dx =
∫ 1
0fαυhdx , ∀υh ∈ Vh
The variational formulation is:
uh ∈ Vh, a(uh, υh) = `(υh) ∀υh ∈ Vh (2)
which, using the representation uh =∑N
j=1 zjφj , can betransformed to the linear system
N∑j=1
zj
∫ 1
0(φ′
iφ′
j )dx =
∫ 1
0fαφidx , 1 ≤ i ≤ N
This system can be rewritten as Au = b, where- u = (z1, . . . , zN)T is the vector of unknown coefficients,- b = (
∫ 10 fφidx , . . . ,
∫ 10 fφN−1dx ,
∫ 10 fφNdx)T is the load
vectorAbiy D. Zeleke and Bahru T. Leyew
Finite Element Convergence Analysis for non-smooth solutions6 / 17
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FEM Continued
The FEM thus becomes:
uh ∈ Vh,
∫ 1
0(u′
hυ′
h)dx =
∫ 1
0fαυhdx , ∀υh ∈ Vh
The variational formulation is:
uh ∈ Vh, a(uh, υh) = `(υh) ∀υh ∈ Vh (2)
which, using the representation uh =∑N
j=1 zjφj , can betransformed to the linear system
N∑j=1
zj
∫ 1
0(φ′
iφ′
j )dx =
∫ 1
0fαφidx , 1 ≤ i ≤ N
This system can be rewritten as Au = b, where- u = (z1, . . . , zN)T is the vector of unknown coefficients,- b = (
∫ 10 fφidx , . . . ,
∫ 10 fφN−1dx ,
∫ 10 fφNdx)T is the load
vectorAbiy D. Zeleke and Bahru T. Leyew
Finite Element Convergence Analysis for non-smooth solutions6 / 17
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FEM Continued
The FEM thus becomes:
uh ∈ Vh,
∫ 1
0(u′
hυ′
h)dx =
∫ 1
0fαυhdx , ∀υh ∈ Vh
The variational formulation is:
uh ∈ Vh, a(uh, υh) = `(υh) ∀υh ∈ Vh (2)
which, using the representation uh =∑N
j=1 zjφj , can betransformed to the linear system
N∑j=1
zj
∫ 1
0(φ′
iφ′
j )dx =
∫ 1
0fαφidx , 1 ≤ i ≤ N
This system can be rewritten as Au = b, where- u = (z1, . . . , zN)T is the vector of unknown coefficients,- b = (
∫ 10 fφidx , . . . ,
∫ 10 fφN−1dx ,
∫ 10 fφNdx)T is the load
vectorAbiy D. Zeleke and Bahru T. Leyew
Finite Element Convergence Analysis for non-smooth solutions6 / 17
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Stiffness Matrix
Entries of the stiffness matrix can be calculated asAij =
∫ 10 (φ
′
iφ′
j )dx ;Using the formulae∫ 1
0(φ′
iφ′
i−1)dx = −1h, 2 ≤ i ≤ N,∫ 1
0(φ′
i )2dx =
2h, 1 ≤ i ≤ N,
We obtain for the stiffness matrix
a(φi , φj) =
2h , i = j ;−1
h , |i − j | = 1;0, otherwise.
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions7 / 17
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Stiffness Matrix
Entries of the stiffness matrix can be calculated asAij =
∫ 10 (φ
′
iφ′
j )dx ;Using the formulae∫ 1
0(φ′
iφ′
i−1)dx = −1h, 2 ≤ i ≤ N,∫ 1
0(φ′
i )2dx =
2h, 1 ≤ i ≤ N,
We obtain for the stiffness matrix
a(φi , φj) =
2h , i = j ;−1
h , |i − j | = 1;0, otherwise.
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions7 / 17
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The Discretization Error
Lemma (Cea)
Let the bilinear form a : H1 × H1 −→ R be continuous andcoercive. Then the solutions u ∈ H1 and uh ∈ H1
h of thecontinuous and discrete variational problems satisfy the relation
‖u − uh‖H1 ≤ C0‖u‖H1 . (3)
Idea of proof.
Use Galerkin Orthogonality a(u − uh, vh) = 0, ∀vh ∈ H1h ,
Coercivity of a, i.e. a(u,u) ≥ γ‖u‖2H1 , ∀u ∈ H1,0 ≤ γ ≤ C ≤ ∞,and continuity of a, i.e. |a(u, v)| ≤ C‖u‖H1‖v‖H1∀u, v ∈ H1
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The Discretization Error (Cont.)
Theorem
Let u ∈ H2(Ω) and uh be the solutions to the continuous anddiscrete variational equations respectively. Then for theinterpolation error satsfies
‖u − uh‖H1(Ω) ≤ C1h‖u‖H2(Ω). (4)
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The Discretization Error (Cont.)
Theorem
Let u ∈ H2(Ω) and uh be the solutions to the continuous anddiscrete variational equations respectively. Then for theinterpolation error satsfies
‖u − uh‖H1(Ω) ≤ C1h‖u‖H2(Ω). (4)
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The Discretization Error (Cont.)
Idea of proof.
Since H1(0,1) ⊂ C[0,1], we can define an interpolationoperator Ih, cont. piecewise linear function, which coincideswith v at nodes as vh(xi) = v(xi).Therefore,
infvh∈Vh ‖u − uh‖H1 ≤ ‖u − Ihu‖H1(Interpolation error).Focus on Interpolation error and to get the convergenceresult use the following steps
1 Localization of the error on subdomains(Partitions),2 Transformation of the subdomains onto the unit interval,3 Calculation of the local Interpolation error,4 Inverse transformation back to subdomains.
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Interpolation theorem
Theorem (An Exact Interpolation Theorem)Let T : X0 + X1 7→ Y0 + Y1 be linear operator with properties
‖Tx‖Y0 ≤ c0‖x‖X0 and ‖Tx‖Y1 ≤ c1‖x‖X1 ,
let be either 0 < θ < 1 and 1 ≤ q <∞ or 0 ≤ θ ≤ 1 and q =∞.Then for Xθ = (X0, X1)θ,q;K and Yθ = (Y0, Y1)θ,q;K
‖Tx‖Yθ≤ c1−θ
0 cθ1‖x‖Xθ. (5)
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The Problem:
Define T : H1 −→ H1 by Tu = u − uh then using equation (3) ,(4) and the interpolation theorem, we have:
‖u − uh‖H1 ≤ C1−θ0 Cθ
1hθ‖u‖H1+θ . (6)
For our model problem the exact solution can be approximatedas:- u(x) ≈ xα+2
∫ 1
0|xα+2|2dx =
12α+5 , α > −5
2 ;∞, else.
∫ 1
0|xα+1|2dx =
12α+3 , α > −3
2 ;∞, else.∫ 1
0|xα|2dx =
12α+1 , α > −1
2 ;∞, else.
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The Problem:
Define T : H1 −→ H1 by Tu = u − uh then using equation (3) ,(4) and the interpolation theorem, we have:
‖u − uh‖H1 ≤ C1−θ0 Cθ
1hθ‖u‖H1+θ . (6)
For our model problem the exact solution can be approximatedas:- u(x) ≈ xα+2
∫ 1
0|xα+2|2dx =
12α+5 , α > −5
2 ;∞, else.
∫ 1
0|xα+1|2dx =
12α+3 , α > −3
2 ;∞, else.∫ 1
0|xα|2dx =
12α+1 , α > −1
2 ;∞, else.
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The K Method
The solution:
xα+2∈ H1, α > −3
2 ;6∈ H2, α ∈ (−3
2 ,−12 ].
Equivalent to: v := u′(x) ≈ xγ for γ = α + 1
xγ∈ L2, α > −3
2 ;6∈ H1, α ∈ (−3
2 ,−12 ].
Now we need to find θ so that xγ ∈ Hθ for γ ∈ (−12 ,
12) . xγ ∈ Hθ
if ∫ ∞0
t−(2θ+1)K (t ; v)2dt <∞
with
K (t ; v)2 = infv=v0+v1
(‖v0‖2L2+ t2‖v1‖2H1
).
We know K (t ; v) ≤ ‖v‖L2 .Abiy D. Zeleke and Bahru T. Leyew
Finite Element Convergence Analysis for non-smooth solutions13 / 17
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The K Method
The solution:
xα+2∈ H1, α > −3
2 ;6∈ H2, α ∈ (−3
2 ,−12 ].
Equivalent to: v := u′(x) ≈ xγ for γ = α + 1
xγ∈ L2, α > −3
2 ;6∈ H1, α ∈ (−3
2 ,−12 ].
Now we need to find θ so that xγ ∈ Hθ for γ ∈ (−12 ,
12) . xγ ∈ Hθ
if ∫ ∞0
t−(2θ+1)K (t ; v)2dt <∞
with
K (t ; v)2 = infv=v0+v1
(‖v0‖2L2+ t2‖v1‖2H1
).
We know K (t ; v) ≤ ‖v‖L2 .Abiy D. Zeleke and Bahru T. Leyew
Finite Element Convergence Analysis for non-smooth solutions13 / 17
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The K Method
The solution:
xα+2∈ H1, α > −3
2 ;6∈ H2, α ∈ (−3
2 ,−12 ].
Equivalent to: v := u′(x) ≈ xγ for γ = α + 1
xγ∈ L2, α > −3
2 ;6∈ H1, α ∈ (−3
2 ,−12 ].
Now we need to find θ so that xγ ∈ Hθ for γ ∈ (−12 ,
12) . xγ ∈ Hθ
if ∫ ∞0
t−(2θ+1)K (t ; v)2dt <∞
with
K (t ; v)2 = infv=v0+v1
(‖v0‖2L2+ t2‖v1‖2H1
).
We know K (t ; v) ≤ ‖v‖L2 .Abiy D. Zeleke and Bahru T. Leyew
Finite Element Convergence Analysis for non-smooth solutions13 / 17
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The K Method
The solution:
xα+2∈ H1, α > −3
2 ;6∈ H2, α ∈ (−3
2 ,−12 ].
Equivalent to: v := u′(x) ≈ xγ for γ = α + 1
xγ∈ L2, α > −3
2 ;6∈ H1, α ∈ (−3
2 ,−12 ].
Now we need to find θ so that xγ ∈ Hθ for γ ∈ (−12 ,
12) . xγ ∈ Hθ
if ∫ ∞0
t−(2θ+1)K (t ; v)2dt <∞
with
K (t ; v)2 = infv=v0+v1
(‖v0‖2L2+ t2‖v1‖2H1
).
We know K (t ; v) ≤ ‖v‖L2 .Abiy D. Zeleke and Bahru T. Leyew
Finite Element Convergence Analysis for non-smooth solutions13 / 17
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K-Method(cont.)
we focus on ∫ 1
0t−(2θ+1)K (t ; v)2dt <∞
Decompose v(x) = v0(x) + v1(x) for ε ∈ (0, 12)
v0(x) = max(xγ − εγ ,0)
v1(x) = min(xγ , εγ)
‖v0‖2L2+ t2‖v1‖2H1
=
∫ 1
0|v0(x)|2dx + t2
∫ 1
0(|v1(x)|2 + |v ′1(x)|2)dx
g(t ; ε) :=2γγ + 1
ε2γ+1 − γ2
2γ − 1t2ε2γ−1 + C
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K-Method(cont.)
we focus on ∫ 1
0t−(2θ+1)K (t ; v)2dt <∞
Decompose v(x) = v0(x) + v1(x) for ε ∈ (0, 12)
v0(x) = max(xγ − εγ ,0)
v1(x) = min(xγ , εγ)
‖v0‖2L2+ t2‖v1‖2H1
=
∫ 1
0|v0(x)|2dx + t2
∫ 1
0(|v1(x)|2 + |v ′1(x)|2)dx
g(t ; ε) :=2γγ + 1
ε2γ+1 − γ2
2γ − 1t2ε2γ−1 + C
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions14 / 17
![Page 34: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/34.jpg)
K-Method(cont.)
Minimize g(t ; ε) over ε.
g′(t ; ε) =
2(2γ + 1)γ
γ + 1ε2γ − γ2t2ε2γ−2 = 0
gives
ε = t
√(γ + 1)γ
2(2γ + 1)= C1t
g′′
(t ; ε) =4(2γ + 1)γ2
γ + 1ε2γ−1 − 2γ2(γ − 1)t2ε2γ−3 ≥ 0.
Therefore
infε
g(t ; ε) =2γ
γ + 1)(C1t)2γ+1 − γ2
2γ − 1t2(C1t)2γ−1 + C = C2t2γ+1
Hence
K (t ; v)2 ≤ C2t2α+3.
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions15 / 17
![Page 35: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/35.jpg)
K-Method(cont.)
Minimize g(t ; ε) over ε.
g′(t ; ε) =
2(2γ + 1)γ
γ + 1ε2γ − γ2t2ε2γ−2 = 0
gives
ε = t
√(γ + 1)γ
2(2γ + 1)= C1t
g′′
(t ; ε) =4(2γ + 1)γ2
γ + 1ε2γ−1 − 2γ2(γ − 1)t2ε2γ−3 ≥ 0.
Therefore
infε
g(t ; ε) =2γ
γ + 1)(C1t)2γ+1 − γ2
2γ − 1t2(C1t)2γ−1 + C = C2t2γ+1
Hence
K (t ; v)2 ≤ C2t2α+3.
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions15 / 17
![Page 36: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/36.jpg)
K-Method(cont.)
Minimize g(t ; ε) over ε.
g′(t ; ε) =
2(2γ + 1)γ
γ + 1ε2γ − γ2t2ε2γ−2 = 0
gives
ε = t
√(γ + 1)γ
2(2γ + 1)= C1t
g′′
(t ; ε) =4(2γ + 1)γ2
γ + 1ε2γ−1 − 2γ2(γ − 1)t2ε2γ−3 ≥ 0.
Therefore
infε
g(t ; ε) =2γ
γ + 1)(C1t)2γ+1 − γ2
2γ − 1t2(C1t)2γ−1 + C = C2t2γ+1
Hence
K (t ; v)2 ≤ C2t2α+3.
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions15 / 17
![Page 37: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/37.jpg)
K-Method(cont.)
Minimize g(t ; ε) over ε.
g′(t ; ε) =
2(2γ + 1)γ
γ + 1ε2γ − γ2t2ε2γ−2 = 0
gives
ε = t
√(γ + 1)γ
2(2γ + 1)= C1t
g′′
(t ; ε) =4(2γ + 1)γ2
γ + 1ε2γ−1 − 2γ2(γ − 1)t2ε2γ−3 ≥ 0.
Therefore
infε
g(t ; ε) =2γ
γ + 1)(C1t)2γ+1 − γ2
2γ − 1t2(C1t)2γ−1 + C = C2t2γ+1
Hence
K (t ; v)2 ≤ C2t2α+3.
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions15 / 17
![Page 38: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/38.jpg)
K-Method(cont.)
Minimize g(t ; ε) over ε.
g′(t ; ε) =
2(2γ + 1)γ
γ + 1ε2γ − γ2t2ε2γ−2 = 0
gives
ε = t
√(γ + 1)γ
2(2γ + 1)= C1t
g′′
(t ; ε) =4(2γ + 1)γ2
γ + 1ε2γ−1 − 2γ2(γ − 1)t2ε2γ−3 ≥ 0.
Therefore
infε
g(t ; ε) =2γ
γ + 1)(C1t)2γ+1 − γ2
2γ − 1t2(C1t)2γ−1 + C = C2t2γ+1
Hence
K (t ; v)2 ≤ C2t2α+3.
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions15 / 17
![Page 39: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/39.jpg)
Application of Interpolation theorem
Then ∫ 1
0t−(2θ+1)K (t ; v)2dt ≤
∫ 1
0t2(α−θ+1)dt <∞,
if 2(α− θ + 1) > −1 =⇒ θ < 32 + α.
This implies
v(x) = xα+1 ∈ Hθ, for θ <32
+ α,
=⇒ u(x) = xα+2 ∈ H52 +α, for α ∈ (−3
2,−1
2).
Then we can estimate the error as h→ 0 in
‖u − uh‖H1 ≤ C1−θ0 Cθ
1hθ‖u‖H1+θ . (7)
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions16 / 17
![Page 40: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/40.jpg)
Application of Interpolation theorem
Then ∫ 1
0t−(2θ+1)K (t ; v)2dt ≤
∫ 1
0t2(α−θ+1)dt <∞,
if 2(α− θ + 1) > −1 =⇒ θ < 32 + α.
This implies
v(x) = xα+1 ∈ Hθ, for θ <32
+ α,
=⇒ u(x) = xα+2 ∈ H52 +α, for α ∈ (−3
2,−1
2).
Then we can estimate the error as h→ 0 in
‖u − uh‖H1 ≤ C1−θ0 Cθ
1hθ‖u‖H1+θ . (7)
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions16 / 17
![Page 41: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/41.jpg)
Application of Interpolation theorem
Then ∫ 1
0t−(2θ+1)K (t ; v)2dt ≤
∫ 1
0t2(α−θ+1)dt <∞,
if 2(α− θ + 1) > −1 =⇒ θ < 32 + α.
This implies
v(x) = xα+1 ∈ Hθ, for θ <32
+ α,
=⇒ u(x) = xα+2 ∈ H52 +α, for α ∈ (−3
2,−1
2).
Then we can estimate the error as h→ 0 in
‖u − uh‖H1 ≤ C1−θ0 Cθ
1hθ‖u‖H1+θ . (7)
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions16 / 17
![Page 42: Finite Element Convergence Analysis for non-smooth solutions](https://reader031.vdocuments.us/reader031/viewer/2022012500/6179ece92cb85c452b0c91a2/html5/thumbnails/42.jpg)
Thank you!
Abiy D. Zeleke and Bahru T. LeyewFinite Element Convergence Analysis for non-smooth solutions17 / 17