asymptotic and numerical analysis of charged particle beamsmohammad/meetings/rgd/beams_sddg.pdf ·...

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Asymptotic and Numerical Analysis of Charged Particle Beams Mohammad Asadzadeh Chalmers University of Technology SE-412 96 G ¨ oteborg, Sweden E-mail: [email protected] URL: http://www.math.chalmers.se/˜mohammad M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.1/20

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Page 1: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Asymptotic and Numerical Analysis of Charged ParticleBeams

Mohammad Asadzadeh

Chalmers University of Technology

SE-412 96 Goteborg, Sweden

E-mail: [email protected]

URL: http://www.math.chalmers.se/˜mohammad

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.1/20

Page 2: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Outline

The continuous problem — Asymptotic analysis

existence, uniqueness, regularityconvergence of solution for VFP to that of VP

iterative scheme, and stabilitiesbasic estimates and canonical representations

The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20

Page 3: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Outline

The continuous problem — Asymptotic analysisexistence, uniqueness, regularity

convergence of solution for VFP to that of VP

iterative scheme, and stabilitiesbasic estimates and canonical representations

The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20

Page 4: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Outline

The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP

iterative scheme, and stabilitiesbasic estimates and canonical representations

The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20

Page 5: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Outline

The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP

iterative scheme,

� �

and

� �

stabilities

basic estimates and canonical representations

The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20

Page 6: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Outline

The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP

iterative scheme,

� �

and

� �

stabilitiesbasic estimates and canonical representations

The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20

Page 7: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Outline

The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP

iterative scheme,

� �

and

� �

stabilitiesbasic estimates and canonical representations

The discrete problem—Numerical Analysis

A scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20

Page 8: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Outline

The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP

iterative scheme,

� �

and

� �

stabilitiesbasic estimates and canonical representations

The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary condition

The standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20

Page 9: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Outline

The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP

iterative scheme,

� �

and

� �

stabilitiesbasic estimates and canonical representations

The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methods

Discontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20

Page 10: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Outline

The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP

iterative scheme,

� �

and

� �

stabilitiesbasic estimates and canonical representations

The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variable �

numerical experiments

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20

Page 11: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Outline

The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP

iterative scheme,

� �

and

� �

stabilitiesbasic estimates and canonical representations

The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variable �numerical experiments

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20

Page 12: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

The Vlasov – Poisson – Fokker – Planck system

�������������������

�� �� ��� � � ��� � � � ��� � � ��� ���� � � � � �! � � � "# �

� ���� � � � � � �%$ ���� � � � ���� � � � � �& � �

� � �� " � � ' �( � � � )

*� � ) * � + � ) � ( )� + � �

� � � ���� � � " � ( �

� � � �,� -� � �� - � .0/ +� - ��� " � 1 ��� as

*�* 1 2

or

� � � �3� �4 - �

where

4 ���65 �

, external potential force,-

internal potential field

Difficulty in solving Cauchy problem in VP:

is singular (up with

(

).Control of

7 + 7 � ensure sufficient regularity for

to construct uniquesolution.

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.3/20

Page 13: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Technical steps

Key issue:

� �

estimates for + and

Difficulty:

�� � �� � � � � Fluid Eqs with too involved� �

estimates

Study: deterministic

Key idea: the maximum principle yields an estimate of� ����� � � �� *� * � �� � � � �

��� � � " � �

for � # (

and by interpolation we get error bounds for

7 + 7 � � 7 ��� + 7 � � 7 � 7� � 7 ��� � 7�

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.4/20

Page 14: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Existence, uniqueness, regularity, convergence

Theorem 0. (VFP) Assume that

�$ 5 ��� �$ � � � � � � � � � � � � � � � � � � �

�� *� * � � � � � * �%$ * *� �%$ * *� � �$ * � � � � � � � � �Then there is a unique solution

� �� � �

to the VFP equation satisfying

�5 � � � � � ���� �� ��� � � � � � � � � � � �

�� *� * � � � � � * � * *� � * � � � ���� � ��� � � � � � � � � � �

� � � ���� � � � � � � � � � � �� � �Theorem 0. (VP) If

� �� � �

and� �� � �� �

are solutions of VP and VFP respectively then� ����$ �� � 7 � � � �� � � " � 7 � 7�� � � � � �� � � " � 7�

7 � � � �� � � " � 7�

� � � � � �

� � � �� *� * � � � � �M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.5/20

Page 15: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Construction of iterative scheme

Assume that

�� ���� " �

is given and is as regular as

in theorem VFP

Solve the Fokker-Planck equation� � � �� � ��� � � � � �� �� � � � � � � ��� � � � � � ���

� � � � ���� � � � � � �$ ���� � �

Compute�����������

+� � � ���� " � � � � � � ��� � � " � ( � �

�� � � ��� " � � ' � ( � � � )

*� � ) * � +� � � � ) � ( )

The above FP equation has a unique solutionLions, the father, degenerate type problems

Iterating we get a solution to VFP.

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.6/20

Page 16: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

The linear (Vlasov) Fokker–Planck equation

���������������

�� �� ��� � �� ��� � � � ��� � � �

� ���� � � � � � �$ ���� � �

�� ���� � � " � � � � � ��� � � " � � �� � �

given

Assumptions

(A1)

�%$ � � � � � � � � � � � � � � ��� � � �� � �� � � � �� � �� � �

(A2)

�� � �� �� � � � � � � � � � � �

Solution space

� � � � � � � � � � � �� � � � � � �� � � ��

�� � �� � � � �

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.7/20

Page 17: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Assume (A1) and (A2), thenFP has a unique solution

� � �

�� � � � � ��� � � � � �

weak solution of

� � � �� � �

and

�� � �

�%$ 5 ��� �5 � � �5 ��� positivity (Tartar)

�$ � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � ��� � � � � �

7 � � " � 7�

� 7 �$ 7�

$7 � ��� � 7

�( � � Maximum principle

Assume that

��

is divergent free, i.e.

� � �� � �

, then

�$ � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � � � � � � � �

7 � � " � 7 � � 7 �$ 7�

�$

7 � �� � 7�

( � � � �

- stability

Idea:

��

: let�� � � � � � �

, then for

# � � 7 ��� �� 7�

� �� �� � � � � � �� �� ���� � � � � � �$ ���� � �

has a unique solution

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.8/20

Page 18: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Standard forms

Omit the superscripts � and � �

. Consider VFP and its differentiatedform in

���� � �

. Recall that

� � � � �� �

and

� �� � �

denote the solutions of VFPand VP, respectively. Let

�� � � � ��

, then

�� �� �� � � �� � � � ��� � � �

�� � � � �� �� �� � � � �,� �� � � � � �� �� � � � � � � �

�� � � �� �� � ��� �� � � �� �� � � � � ��� � � � � � �� � � ��� ��

The second equation is a vector equation with

� � � � � � �� �3� � � �

is a

�( �(

matrix decomposed in( (

blocks:

� �� �(

��� � � �

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.9/20

Page 19: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Canonical form

Multiplying standard forms by � � � �� *� * � �� � �

to obtain

� � �� ��� � � � � � �� *� * � � ��� � � � ��� � ��

� � �

� ��

�������������������������������

�� � � � �� *� * � � � � � �

� �� � � � � � � �� *� * � � �� � � � � � �

� �� � � � � � � � *� * �� *� * � � � � � � � � � � �

� ��� � � �� � �

���������������������������������������

�� � � �� *� * � �� � �� �

� � � � ��

�� � �

� � � � �� �� � �

� � � � �� *� * � � � � � � � �

� � � �

�� � � � �� *� * � �� � � � � � �� �

� �� � � ��

� � � �� �

� �� � � �� � � � �� �

� � � � � � �� *� * � �� � � ��� ��

� �� � � � � � �� � �� *� * � � � � � �3� �

Estimate each

� �� , for (

� �� � �

) and use Grönwall type inequalities, ...

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.10/20

Page 20: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

The final form of pencil beam equations

The Fokker-Planck equation�����������������������

��� �3� � � � �� � � �� � ���

� ����

�� � � � � �� �

�� � � �

� �� �

� � �� � � �

� � � � )� ��� � � �� � � � � ) � � � � � � � � � � �� / � � � � � � �

� � �� )� �� � � �� � � ��� � � � � � ���Forward peakedness allow: to project the FP operator from acting onthe right half of the unit sphere

� �

into the tangent plane at the point�� � ��� � �

to

� �

. In this way we get

the Fermi equation:�������

� ��

�� �

� ) � �

� � �� � � �

� �� � � � �� � � � ��� � � � � �� � � � � �

� � � � )� ��� �� � � � � ) � � � ) � � � � � � � � � with

� �� � � � ��

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.11/20

Page 21: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Pencil beam models in two (space) dimensions

In 2 space dimensions; we introduce the current

�� � �� � �

, the scalingvariable �� � � �� � � � � / � � � / � � �

, and the scaled forward current

�� � � � ��� )� � �� � � � � � �� � � � ,For this

, we get the canonical form of pencil beam equations:�������

� ��

�� �

� ) � � � �� ��

�� ��� �

� � ��� )� � � � � � ) � � � � � � )� � � ��

The operator

for the Fokker-Planck equation is:�������

� ��

� �� � � �

�� �

�� � � � � � �

� � � � �� � � � � � � � � � �� � � � � � � �

whereas the corresponding Fermi equation has simply

� �� �

� � � �

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.12/20

Page 22: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Numerical domain/Canonical form of a Fermi model

Fermi equation, in a slab of thickness

, � � �� � � � ��� � �, with symmetric

cross section

��� � � ��� ��� � � � � ) $ � ) $ � � � � $ � � $ � :��������

��������� � �� � �� � � in

� � �� �� �

��

���� )� � � $ � � ��� for

���� ) � � �� �� �

� � � � � � � � � �� � � � for � � � �� �

� � ��� � � � � ��� on

� ��� � � ���� � � � � �� � � � � � � �� � � � �

�� � �� � ��� � �

and

� � is the outward unit normal to

at

��� � � � � �

.

This equation is interpreted as:

time-dependent ( � viewed as time variable),

degenerate (convection in ), diffusion in �),

forward-backward ( � changes sign),

convection dominating ( is small),

convection-diffusion problem.

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.13/20

Page 23: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

The (numerical) phase-space domain

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

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� � � � � � � � � � � � � � � � � � � � � � � �

z

y

x

u = 0

u = 0

z

z

n=(0,−1,0)

n=(0,1,0)

z

y0

0

u(x,y,z)=00

0−

u(0,y,z)=f(y,z)

u(x,y,z)=0

Figure 0: 2D-ModelM. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.14/20

Page 24: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Fully discrete strategy

For convection dominated problems, having hyperbolic nature, thestandard Galerkin (SG) method converges with the rate

� �� � �, (versus

� �� � � � �

for elliptic and parabolic problems), provided that the exactsolution is in the Sobolev space

� � � �

.To speed up the convergence of SG we introduce the semi-streamlinediffusion (SSD) method, through a modified form of the test functions.This add (automatically) a proper amount of viscosity resulting insmoothing effects on the equation.SSD method is performed only on the � � variable, whereas in the usualstreamline-diffusion (SD) method both � � and � discretizations areperformed is one, and the same, single variational formulation.In our approach, however, the penetration variable � is interpreted as atime variable and discretized by: discontinuous Galerkin (DG), backwardEuler (BE) and Crank-Nicolson (CN) methods.In fully discrete problem we combine SG or SSD schemes for

� � with atime discretization method for the penetration interval

�� .

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.15/20

Page 25: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

The Standard Galerkin Method

This is a finite element approximation, in � � � � )� � � , based onquasi-uniform triangulation of

� � � ��� ��� , with a mesh size

�:

� � � � �

.To this approach we define the inflow boundary

� � � � �� � � �� � � �� � � � � � � � � � � � � �� � �

and a discrete, finite dimensional, function space��� � � �

� �� �

with

� � � ��� � � �� � � � � �� �� � � �

on

� �� �

such that,

�� � � �

� ��� � � �� � ��� �

,

� � �

� � ��� 7� � � 7�� � '� � � � 7� 7� � � � � � � and

� � � � � �

An example of such

� � is the set of sufficiently smooth piecewisepolynomials

� � � � �

of degree� �, satisfying the boundary conditions.

Now the objective is to find� � � ��� , such that� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ��� �

� � � ��� � � � � � � �� � �

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.16/20

Page 26: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

A Smoothing Petrov Galerkin Method

Here we introduce the SSD approach which includes a diffusiongenerating test function in the ) direction over the usual SG procedure.

Using SSD we obtain a non-degenerate type equation with somewhatimproved regularity in the ) direction:

We let� � � � � with

� � �� � �

. Then the SSD test functions: � ��

automatically add the extra diffusion term,� �� � � �

, to the variationalformulation which, combined with

�� � � � � �� � � � � � � ��

, gives anon-degenerate equation with a diffusion term of order

� � � , for

�5 :

Multiplying the differential equation by� �� , integrating over

� � , andusing the boundary conditions yields,

� �� � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � �

� � � � �� � �� ��

� � � ��

The discrete version is now obtained by replacing

by a suitable

� � .M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.17/20

Page 27: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Numerical Implementation

We test convergence of SG and SSD through some numerical examples.Our implementations are performed over four different initial data:modified Dirac, hyperbolic, Maxwellian, and cone functions, approximatingour data: the

-function. The procedure is split into two steps:

Discretize the two dimensional domain

� � � �� ��� by means of

continuous piecewise linears: � � �� �

, and establish a mesh there inorder to obtain a semidiscrete solution.

Apply one of the time discretization methods (BE, CN or DG), to stepadvance in � direction.

The error � � � � � � � � , is measured in the weighted

� � norm

* *�� * * ��� � � ��

��* � *

�� � �

�� �� � � � � � � � � ��

with

� � � denoting the midpoints of the edges of the mesh triangles � .

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.18/20

Page 28: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Convergence results/tables

The reference domain is

� � � � ��� � � � � � � � � �

Parameters:

� � ��

� � � � �

�� �

(for discretization in �), and� � � ��

� � �

.SG in � � � � )� � � extrapolation error �� � � � �� � � �

discrete � Dirac Hyperbolic Maxwellian Cone

BE 13.63-1.806 .064-.013 .123-.042 .115-.047CN 13.73-1.814 .065-.014 .122-.041 .115-.047DG 13.40-2.065 .064-.012 .117-.043 .110-.051

SSD in � � � � )� � � , extrapolation error �� � � � �� � � �

discrete � Dirac Hyperbolic Maxwellian Cone

BE 13.33-1.801 .063-.014 .118-.041 .110-.045CN 13.44-1.806 .063-.015 .117-.040 .110-.045DG 13.28-2.068 .063-.014 .117-.042 .110-.049

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.19/20

Page 29: Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf · The standard Galerkin (SG) and semi–streamline diffusion (SSD) finite element

Reliability of Asymptotic Expansions

Dose intensity (amount of deposited energy per unit volume, per unit time)radiating an elliptic target at the collision site � � �

��

and with � � � � ��

:

M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.20/20