Geiser J. Decomposition Methods for Differential Equations: Theory and Applications

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Discretization Methods 249

B.3.4.1 Sobolev Spaces

For the variational formulation, the application of weak derivates is impor-

tant to have the solution spaces. We introduce the underlying spaces, Sobolev

spaces, which is an application of the Lebesgue norms and spaces to include

the weak derivatives.

We have the following deﬁnition of the weak derivatives.

DEFINITION B.20 Let k be a nonnegative integer, and let f ∈

loc

(Ω). We assume the existence of weak derivatives D

(f) for all |α|≤k.

We deﬁne the Sobolev norm:

||f||

(Ω)

⎛

⎝



|α|≤k

||D

f||

(Ω)

⎞

⎠

1/p

. (B.106)

We deﬁne the Sobolev spaces via

(Ω) :=

⎧

⎨

⎩



|α|≤k

||D

f||

(Ω)

⎫

⎬

⎭

. (B.107)

THEOREM B.3

The Sobolev space W

(Ω) is a Banach space.

THEOREM B.4

Let Ω be any open set. Then C

∞

(Ω) ∪W

(Ω) is dense in W

(Ω) for p<∞.

We have the inclusion of Sobolev spaces.

PROPOSITION B.1

Let Ω be any open set. Then C

∞

(Ω) ∪W

(Ω) is dense in W

(Ω) for p<∞.

In the following we discuss the application of the weak formulation, as the

variational principle, also known as Ritz-Galerkin approach, see [65] and [187].

B.3.4.2 Ritz-Galerkin Approach

The idea is to ﬁnd an unique solution of the weak formulation within a

(ﬁnite-dimensional) subspace.

Therefore, the unique representation of the linear functional is important.

We formulate the Riesz Representation Theorem.

250Decomposition Methods for Diﬀerential Equations Theory and Applications

THEOREM B.5

We can represent any continuous linear functional F on a Hilbert space H

F (v)=(u, v), (B.108)

for some u ∈ H, and we have

||F ||



= ||u||

. (B.109)

PROOF

The uniqueness can be followed by the proof in [29].

Further we suppose the following conditions:

H.1. (H, (·, ·)) is a Hilbert space.

H.2.Vis a (closed) subspace of H.

H.3.a(·, ·) is a bounded, symmetric bilinear form that is coercive on V .

The Ritz-Galerkin approach is the following.

Given a ﬁnite-dimensional subspace V

⊂ V and F ∈ V



, ﬁnd u

∈ V

such

that

a(u

,v)=F (v) ∀v ∈ V

. (B.110)

We can follow the uniqueness of our formulation B.110

THEOREM B.6

Under the conditions H.1.–H.3., there exists a unique u

that solves (B.110).

PROOF

While (V

,a(·, ·)) is a Hilbert space and F |

∈ V



, we can apply the Riesz

Representation Theorem and obtain uniqueness.

In the following we discuss the error estimates.

B.3.4.3 Error Estimates

For the convergence of a Galerkin method we could use the Cea’s lemma.

THEOREM B.7

Let a(·, ·) be a continuous, V-elliptic bilinear form, and for all f ∈ V



we could ﬁnd an unique solution by the Riesz formulation. Then we could

Discretization Methods 251

estimate

||u − u

≤

inf

∈V

||u − v

, (B.111)

where V = H

and H

is a Hilbert space.

PROOF

An unique solution for (B.150) is based on the Lax-Milgram lemma, so we

could use the behavior also for V

⊂ V = H

We have a(u, v

)=f(v

), ∀ v

∈ V

,thusalsoa(u−u

)=0, ∀ v

∈ V

and a(u −u

)=0.

Therefore a(u − u

,u− u

)=a(u − u

,u− v

), ∀ v

∈ V

Then we have with continuity and V-ellipticity: γ ||u − u

≤ M ||u −

||||u − v

||.

REMARK B.14 The partial spaces V

⊂ V are chosen so that they

are asymptotically dense in V .

So each element u ∈ V has a good approximation by the elements v ∈ V

,if

h>0 is suﬃciently small and we have the convergence (B.150): lim

h→0

||u −

|| = 0, and we have the approximation order: inf

∈V

||u − v

||.

The error boundary of the inﬁmum is estimated by an interpolation operator

inf

v∈V

||u − v|| ≤ ||u − Π

u|| (B.112)

with regularity we could estimate for special test spaces.

An overview to the variational formulation with ﬁnite discretization meth-

ods is given in Figure B.2.

B.3.4.4 Construction and Design of Finite Element Methods

We deal with the approximation of the variational problem.

Find u ∈ V such that

a(u, v)=F (v) , ∀v ∈ V, (B.113)

where V is a Hilbert space, F ∈ V



and a(·, ·) is a continuous, coercive bilinear

form.

For such a problem, we can construct ﬁnite-dimensional subspaces V

⊂ V

in a systematic, practical way, see [44].

We deal with three questions:

252Decomposition Methods for Diﬀerential Equations Theory and Applications

a(u ,v ) = f(v )

hh h

for all v in V

a(u,v) = f(v)

for all v in V

sum a( , ) s = f( )

i = 1 .. N

j = 1 .. N

Variational Equation

Galerkin Equations

Galerkin Method

equivalence

i = 1 .. N

j = 1 .. N

for all v in V

J(v) −> min

J(v ) −> min

sum a( , ) s = f( )

Variational Equation

Petrov−Galerkin Method

for all v in V

FIGURE B.2: Variational formulations with ﬁnite discretizations

methods.

1. Simple trial and test space for approximating the operators and func-

tions in the subinterval.

2. A good determination of the function in the subinterval.

3. A transition for the trial function to save the global behavior.

Construction Criteria for the Finite Element Methods

1) Triangulation of the continuous domain

DEFINITION B.21 Let

(i) K ⊂ R

be a bounded closed set with nonempty interior and piecewise

smooth boundary (the element domain),

(ii) P be a ﬁnite dimensional space of functions on K (the space of shape

functions), and

(iii) N = {N

,...,N

} be a basis for P



(the set of nodal variables).

Then (K, P, N) is called a ﬁnite element.

2) Deﬁnition of a test and trial space:

A linear test space is given as

(x)=

⎧

⎨

⎩

(x − x

) ,x∈ Ω

i+1

− x) ,x∈ Ω

i+1

0else

⎫

⎬

⎭

, (B.114)

Discretization Methods 253

for one element.

3) Local and global characteristics:

Here we deal with conform elements. The discrete problem is to embed into

the continuous problem and we obtain: V

⊂ V .

For example H

(Ω) and H

(Ω) in the second-order elliptic problem and we

obtain the following implication, see Lemma B.1.

LEMMA B.1

Let z :

Ω → R be a function, with z|

∈ C

(Ω

),withi =

1,...,N. Then we have the implication

z ∈ C(

Ω) → z ∈ H

(Ω). (B.115)

REMARK B.15

The discretization errors can be classiﬁed into two groups:

1. Approximation error from the approximation of the operators

and functions

2. Approximation error from the numerical integration

The ﬁrst error is estimated by the consistency and stability.

The second error is an error of the approximate integration. Using polyno-

mials we could construct exact integration formulas (e.g., Gauss quadrature

formulas).

Triangular Finite Elements

Let K be any triangle. Let P

denote the set of all polynomials in two

variables of degree less or equal k. We have the following dimensions for the

elements.

The Lagrange element is given in the following example.

For (k =1),letP = P

∞

.LetN

= {N

} (dimP

=3)where

(v)=v(z

), (B.116)

and z

, z

are the vertices of K.

Further we can deﬁne the local interpolant for each ﬁnite element.

DEFINITION B.22 Given a ﬁnite element (K, P, N),lettheset

{φ

:1≤ i ≤ k}⊂Pbe the dual basis to N.Ifv is a function for which all

∈N, i =1,...,k , are deﬁned, then we deﬁne the local interpolant by

v :=



i=1

(v)φ

. (B.117)

254Decomposition Methods for Diﬀerential Equations Theory and Applications

Example: Let K be any triangle.

f = N

(f)(1 − x − y)+N

(f)x + N

(f)y (B.118)

=1. (B.119)

B.3.4.5 Theory of Conforming Finite Element Methods

The conforming theory of ﬁnite element methods is based on the Ritz

method for symmetric and on the Galerkin method for the nonsymmetric

bilinear forms.

Important is the embedding of the ﬁnite space into the inﬁnite or continuous

space: V

⊂ V .

We could formulate Theorem B.8.

THEOREM B.8

Because of V

⊂ V and with the same scalar product (·, ·) we could deﬁne a

Hilbert space and a(·, ·) has for V

the same behavior as for V .

Then there exists an unique solution u

∈ V

with the suﬃcient optimality

criteria:

a(u

)=f(v

) for all v

∈ V

. (B.120)

This approximative treatment is used as the Ritz method. For unsymmet-

ric bilinear forms the discrete variational formulation is called the Galerkin

method.

Error Estimates for the Conforming Methods

For the convergence of a Galerkin method we could use the Cea’s lemma.

THEOREM B.9

Let a(·, ·) be a continuous, V-elliptic bilinear form, such that for all f ∈ V

∗

we could ﬁnd a unique solution by the Riesz formulation. Then we could

estimate

||u − u

|| ≤

inf

∈V

||u − v

||. (B.121)

PROOF

The unique solution for (B.150) is based on the Lax-Milgram lemma, so we

could use the behavior also for the subspace V

⊂ V .

We have

a(u, v

)=F (v

) ∀ v

∈ V

, (B.122)

Discretization Methods 255

so also,

a(u − u

)=0∀ v

∈ V

, (B.123)

and

a(u − u

)=0. (B.124)

Therefore, we have

a(u − u

,u− u

)=a(u − u

,u− v

) ∀ v

∈ V

. (B.125)

Then we have with continuity and V-ellipticity

γ ||u − u

≤ M ||u − u

||||u − v

||. (B.126)

Regularity for Solution and Embedding of Solutions to Strong

Solutions

The idea is to embed the classical solution into the weak solutions with the

boundary conditions.

REMARK B.16 The regularity of the solution and right-hand side

is not enough, the regularity of the domain is also necessary.

The problem of the regularity can be shown in the following example.

Example B.3

We deal with a domain, including a discontinuous edge, see Figure B.3.

Discontinuous Edge

FIGURE B.3: Domain with Discontinuous Edge.

We have the analytical solution, see [27], given as

u(r, φ)=r

π/(2ω)

sin(

2ω

φ). (B.127)

256Decomposition Methods for Diﬀerential Equations Theory and Applications

For example, with ω = π, we have a singularity in r → 0 such that u/∈ H

(Ω).

So we do not have a strong solution because of the discontinuity of the

singular point r → 0.

In the following we present the embedding to obtain continuity of the

solutions.

THEOREM B.10

The embedding formula for the continuity of the solution (e.g., weak solutions

are embedded in strong solutions), is given as

Continuous embedding:

We have f ∈ H

(Ω) with k>n/2. Then we have for the solution,

u ∈ C

(Ω) ∩ H

(Ω), (B.128)

and u is a classical solution.

PROOF

Because of u ∈ H

k+2

∩H

,wehaveu ∈ H

k+1

(Ω) and with the continuous

embedding W

k+2

(Ω) → C

(Ω) for k>n/2wegettheassumption.

B.3.4.6 Interpolant (Reference Element), Theory of Conforming

Elements

The interpolation property could be studied more simply by introducing a

reference discretization.

LEMMA B.2

Given is a partial domain K and the reference element K



with an

aﬃne translation:

x = F (p)



= Bp + b for p ∈ K



. (B.129)

Then we have the following properties:

i) u ∈ H

(K) ↔ v ∈ H



ii) |v|

l,K



≤ c||B||(detB)

−1/2

|u|

l,K

∈ H

(K).

Example: quasi-uniform triangulation:

|v|

r,K



≤ c{ inf

p∈K



s(p)

−1/2

|u|

r,K

. (B.130)

Discretization Methods 257

Interpolation Error of the Conforming Theory

In the conforming theory the interpolation theory is important.

The idea is to ﬁnd a “good” projection operator:

||u − u

|| ≤ c inf

v∈V

||u − v|| ≤ c||u − Π

u||.

We have the following theorem:

THEOREM B.11

We have the piecewise deﬁned projector over the triangulation Z and for

polynomials of order k.Forr ≤ k and regularity conditions for the reference

element we could denote

||u − Π

u||

≤ ch

k+1−r

||u||

k+1

, (B.131)

where h>0 assigns the ﬁneness of the triangulation Z.

Theory of the Conforming Elements to Nonlinear Boundary Value

Problems

We could enlarge our theory to nonlinear boundary value problems. The

idea is to describe the monotone operators.

Important are the monotony and the start position of the iteration.

We could transfer the linear results to the nonlinear results, if we have the

following characterization:

• a) It exists a γ>0with:(Bu −Bv, u −v) ≥ γ||u −v||

for all u, v ∈ V

(we call the operator to be strong monotone).

• b) It exists a M>0with:||Bu −Bv|| ≤ M ||u − v|| for all u, v ∈ V .

We treat the abstract operator equation:

(Bu,v) = 0 for all v ∈ V. (B.132)

A generalization of the Lax-Milgram lemma is given in the following lemma

(idea: ﬁxed point iteration).

LEMMA B.3

With respect to a) and b), the operator equation Bu =0hasan

unique solution u ∈ V ,whichistheﬁxedpoint:

v := v − rBv , for all v ∈ V, (B.133)

with T

: V → V for the parameter values r ∈ (0,

2γ

258Decomposition Methods for Diﬀerential Equations Theory and Applications

In the next section we discuss the nonconforming theory of the ﬁnite element

method, which can help to reduce the complexity of the formulation by an

approximation.

B.3.4.7 Nonconforming Theory

The motivation for nonconforming elements is given by

1) The choice of the function spaces V

⊂ V is too complex, for example

with a diﬀerential equation of higher order, we have V

⊂ V .

2) The bilinear form and the linear functional are only approximately com-

putable (e.g., with quadrature formulas).

3) Inhomogeneous, natural boundary conditions or curved boundaries of

the domain Ω do not permit an exact projection of the boundary con-

ditions.

The nonconforming ﬁnite element methods are ﬁnite element methods that

do not directly discretize:

a(u, v)=f(v) for all v ∈ V, (B.134)

with V

⊂ V .

We can formulate our ﬁnite-dimensional problem as

)=f

) for all v

∈ V

, (B.135)

where a

: V

× V

→ R a continuous bilinear form with V

-ellipticity, see

˜γ||v

≤ a

) for all v

∈ V

. (B.136)

With the case V

⊂ V , the bilinear form a

and the linear functional f

are

not certainly deﬁned on V .

So we use the spaces Z

and Z with V → Z and V

→ Z

→ Z with a

(·, ·)

and f

(·) being deﬁned on Z

× Z

and Z

Here we obtain the second Strang lemma.

LEMMA B.4

Second Lemma of Strang, see [186]:

||u − u

|| ≤ c inf

∈V

{||u − z

||+ ||f

− a

, ·)||

∗,h

}. (B.137)

Approximation of the Operators and Functionals