on nite element schemes for vlasov-maxwell...

Thesis for the Degree of Doctor of Philosophy

On finite element schemes forVlasov-Maxwell system and Schrodinger

equation

Christoffer Standar

Division of MathematicsDepartment of Mathematical Sciences

Chalmers University of Technologyand University of Gothenburg

Goteborg, Sweden, 2017

On finite element schemes for Vlasov-Maxwell system and Schrodinger

equation

Christoffer Standar

c© Christoffer Standar, 2017.

Department of Mathematical SciencesChalmers University of Technologyand University of GothenburgSE–412 96 GoteborgSwedenPhone: +46 (0)31-772 10 00

Typeset with LATEX.Printed in Goteborg, Sweden, 2017.

Abstract

This thesis treats finite element schemes for two kind of problems,the Valsov-Maxwell system and the nonlinear Schrodinger equation. Westudy streamline diffusion schemes applied for numerical solution of theone and one-half dimensional relativistic Vlasov-Maxwell system. Thestudy is made both in a priori and a posteriori settings. In the a priorisetting we derive stability estimates and prove optimal convergence rates,due to the maximal available regularity of the exact solution. In additionto this we also prove existence and uniqueness of the numerical solution.In the a posteriori setting we use dual problems to prove error estimates inL∞(H−1) norm. For the Maxwell equation we also prove error estimatesin H−1(H−1) norms. Further more we study a hp-version of the stream-line diffusion scheme for the three dimensional Vlasov-Maxwell system inan a priori setting. A Nitsche type scheme is also introduced and analyzedfor Maxwell’s equations. For the nonlinear Schrodinger equation a twolevel time discretization is used. Here we derive a priori error estimatesboth in L2 and H1 norms.

Keywords: Vlasov-Maxwell, streamline diffusion, a priori error anal-ysis, a posteriori error analysis, hp-scheme, Nitsche scheme, nonlinearSchrodinger equation

i

List of included papers

The following papers are included in this thesis:

• Paper I. Christoffer Standar. On Streamline Diffusion schemesfor the one and one-half Dimensional Relativistic Vlasov-Maxwellsystem, Calcolo, 53 (2016), no. 2, p. 147-169.

• Paper II. Mohammad Asadzadeh and Christoffer Standar. Aposteriori error estimates for the one and one-half DimensionalRelativistic Vlasov-Maxwell system. To appear in BIT NumericalMathematics 2017.

• Paper III. Mohammad Asadzadeh, Piotr Kowalczyk andChristoffer Standar. On hp-Streamline Diffusion and Nitscheschemes for the Relativistic Vlasov-Maxwell System. Preprint.

• Paper IV. Mohammad Asadzadeh and Christoffer Standar.Approximating the nonlinear Schrodinger equation by a two levellinearly implicit finite element method. Preprint.

The following papers are not included in this thesis:

• Mohammad Asadzadeh, Larisa Beilina, Muhammad Naseer andChristoffer Standar. A priori error estimates and computationalstudies for a Fermi pencil-beam equation, arXiv:1606.05085, 2016.

• Mohammad Asadzadeh, Larisa Beilina, Muhammad Naseer andChristoffer Standar. Finite element schemes for Fermi equation, AIPConference Proceedings 1863, 370007 (2017).

• Mohammad Asadzadeh, Piotr Kowalczyk and Christoffer Standar.Convergence of Streamline Diffusion and Nitsche methods for theVlasov-Maxwell System. To appear in JCTT, proceeding of ICTT 25,(2017), Monterey, CA.

iii

Contributions. The author of this thesis has contributed in thefollowing manner to the appended papers:

Paper I and II is mainly the work of the author.

In Paper III the author developed and analyzed the Nitsche scheme.

In Paper IV the author constructed and analyzed the time discretiza-tion scheme.

Acknowledgments

I would like to thank my supervisor Mohammad Asadzadeh for hissupport and encouragement. Also a big thank you to my fellow Ph. D.students at the department. Lastly, I would like to thank my parentsTorgny and Paola and my sister Christine.

Christoffer StandarGoteborg, 2017

v

Contents

Abstract i

List of included papers iii

Acknowledgments v

Introduction 11. Vlasov-type systems 12. The Vlasov-Maxwell system 33. The Schrodinger equation 44. The streamline diffusion method 55. The hp-version 76. Nitsche’s method 77. Summary of papers I-III 88. Summary of paper IV 15References 17

Papers 21

Paper I 231. Introduction 252. Assumptions and notations 283. The Maxwell equations 294. The Vlasov-Maxwell equations 365. SD-based DG for the Maxwell equations 426. SD-based DG for the Vlasov-Maxwell equations 447. Conclusion 46Acknowledgment 46References 47

Paper II 491. Introduction 512. Assumptions and notations 553. A posteriori error estimates for the Maxwell equations 57

vii

viii CONTENTS

4. A posteriori error estimates for the Vlasov equation 675. Conclusions and future works 72References 73

Paper III 771. Introduction 792. Notation and Assumptions 823. hp-Streamline Diffusion Method 844. Stability and Convergence of hp-SDFEM 885. Nitsche’s method for Maxwell equations 986. Numerical Results 1087. Conclusion 110Acknowledgment 112References 112

Paper IV 1151. Introduction 1172. Notation and preliminaries 1193. The spatial discretization Scheme 1214. A time discretization scheme 1255. Convergence Analysis 1286. Concluding remarks 133References 133

Introduction

This thesis work is on construction and analysis of numerical schemes,mostly finite elements, for approximate solution of i) the Vlasov-Maxwellsystem in different dimensions and ii) the nonlinear Schrodinger equa-tion. The study of the Vlasov-Maxwell system concerns different Galerkinschemes, e.g., streamline diffusion and Nitsche, while for the nonlinearSchrodinger equation a time-splitting approach is comined with a back-ground spatial discretization. Below we first give a description of theequations in the thesis. Then, we introduce the different Galerkin schemesthat has been considered. Finally, we present the results of the thesis pa-pers.

1. Vlasov-type systems

The Vlasov equation is a hyperbolic type partial differential equation,which describes the time evolution of the phase-space-time distributionfunction for plasma of charged particles. The equation was first suggestedby Anatoly Vlasov in 1938. The Vlasov equation is often coupled withother equations, to obtain system of equations as mathematical models ofcharge particle transport. The coupling terms are of crucial importance indescribing certain physical phenomenon, e.g., the Vlasov-Maxwell system(VM), which is the focus of our studies in this thesis, describes a self-consistent collisionless plasma under the influence of an electromagneticfield (Vlasov-Poisson is an approximation of the Vlasov-Maxwell systemwith a Poisson equation describing a potential field). The VM systemhas a wide range of applications in plasma physics such as lasers, fusiondevices and particle accelerators.

Among all Vlasov-type systems, the Vlasov-Poisson system is the onethat has been widely studied both in analytical and numerical works.Classical analytical approaches to the Vlasov-Poisson system have beenbased on Schauder’s fixed point theorem which was introduced in a work

1

Vlasov-type systems

on the existence of a classical solution to the two dimensional Vlasov-Poisson system by Ukai and Okabe [37]. Another analytic study is due toBardos and Degond [8] where the global existence of solutions of the threedimensional Vlasov-Poisson system with small data has been proved. Inearly 90s Lions and Perthame [28] and Pfaffelmoser [33] both provedglobal existence of the three dimensional Vlasov-Poisson system in in-dependent works. Then, as part of a survey study of classical kineticmodels, DiPerna and Lions [15] proved the global existence of a weaksolution to the Vlasov-Maxwell system in three dimensions. More elabo-rate analytic study of the VM system is due to Glassey and Schaeffer inseveral settings and various geometries; see, e.g., [18], [19], [21] or [22].In [18] Glassey and Schaeffer proved global existence of classical solutionwith the assumption of almost neutral initial data. In [19] and [21] theyhave studied the global existence with small initial data for VM in oneand one-half dimensions (one space variable and two velocity variables)and two and one-half dimensions (two space variables and three velocityvariables), respectively. We also mention some other relevant studies e.g.,[14] and [30]. In [14] the one and one-half dimensional Vlasov-Maxwellsystem is studied in an interval, while [30] considers the three dimensionalVlasov-Poisson equation with point charges.

As for numerical approximation, the particle methods have long beenvery popular for discretizing Vlasov-type systems. For studies on particlemethods related to the Vlasov equation; see, e.g., [13], [16], [38] or [20].The first three articles concern particle methods for the Vlasov-Poissonsystem in different settings, while in the fourth one the authors study theone and one-half dimensional Vlasov-Maxwell system. Recently, the dis-continuous Galerkin finite element method has become increasingly pop-ular for approximating Vlasov type systems. A discontinuous Galerkinmethod for the VM system with different choices of the numerical flux hasbeen studied by Cheng et al. in [12]. In a series of articles [26], [29], [31]the authors have considered the study of several finite element schemes(including discontinuous Galerkin) for approximating the Maxwell equa-tions. As for a combined system, e.g., in [5] the authors construct andanalyze a streamline diffusion based DG scheme for the Vlasov-Poisson-Fokker-Planck system.

Our objective is to construct a numerical scheme for the Vlasov-Maxwell system that is both reliable (being highly stable) and efficient

2

The Vlasov-Maxwell system

(converges optimally in an L2-based energy norm due to maximal avail-able regularity of the exact solution). A desirable scheme should alsocontain a self-adaptivity mechanism that is capable to circumvent over-refinements.

2. The Vlasov-Maxwell system

The main study in this thesis is the relativistic Vlasov-Maxwell systemdefined by

(2.1)

∂tf + v · ∇xf + q(E + c−1v ×B) · ∇vf = 0,

∂tE = c∇×B − j, ∇ · E = ρ,

∂tB = −c∇× E, ∇ ·B = 0.

Here f is density, in phase-space-time, of particles with mass m, chargeq and velocity

v = (m2 + c−2|v|2)−1/2v (v is momentum).

Further, the charge and current densities are given by

ρ(t, x) = 4π

∫qf dv and j(t, x) = 4π

∫qf v dv,

respectively. The functions E and B describe the electric and magneticfields, respectively.

For simplicity, we choose to describe the relativistic Vlasov-Maxwellsystem in the one and one-half dimensional setting (x ∈ R, v ∈ R2), givenby the following system of equations

(2.2)

∂tf + v1∂xf + (E1 + v2B)∂v1f + (E2 − v1B)∂v2f = 0,

∂xE1 =

∫fdv − ρb(x) = ρ(t, x),

∂tE1 = −∫v1fdv = −j1(t, x),

∂tE2 + ∂xB = −∫v2fdv = −j2(t, x),

∂tB + ∂xE2 = 0.

Considering such geometry, we can derive formulas of d’Alembert type forthe field functions, which facilitate the estimation of the field functions.Studies of higher dimensional models require different treatments. To

3

The Schrodinger equation

proceed we set

M1 =

0 0 01 0 00 1 00 0 1

, M2 =

1 0 00 0 00 0 10 1 0

and let W = (E1, E2, B) and b = (ρ,−j1,−j2, 0). Then, introducing thenotations

G(f) = (v1, E1 + v2B,E2 − v1B)

and∇f = (∂xf, ∂v1f, ∂v2f),

we can write the Vlasov-Maxwell system in a more compact form as

(2.3)

∂tf +G(f) · ∇f = 0,M1∂tW +M2∂xW = b.

3. The Schrodinger equation

The Schrodinger equation is widely used in quantum mechanics andits solution describes molecular, atomic, subatomic as well as macroscopicsystems. In its most general form Schrodinger equation can be written as

i~∂

∂tΨ(t, x) = HΨ(t, x),

where H is the Hamiltonian operator for the system. For the case of asingle particle moving in an electric field the Hamiltonian is given by

H = − ~2

2µ∆ + V (t, x)

and the Schrodinger equation takes the form

(3.1) i~∂

∂tΨ(t, x) = − ~2

2µ∆Ψ(t, x) + V (t, x)Ψ(t, x),

where µ is the reduced mass of the particle and V (t, x) is its potentialenergy. In Paper IV we study a nonlinear version of (3.1). Let f :[0,∞) → R and set, for simplicity, all constants to be equal to 1. Thenthe nonlinear Schrodinger equation (NLS) can be written as

(3.2) ut = i∆u+ if(|u|2)u+ g(t, x).

For a real λ and with f(x) = λx the above equation is known as the cubicSchrodinger equation, with eigenvalue λ.

4

The streamline diffusion method

Some analytical results on the NLS equation are discussed in a workby Strauss [35]. The nonlinearity considered in [35] is of the form f(x) =

λx(p−1)/2 for p > 1. Under specific conditions on λ and p there existsa unique global solution to the NLS equation. Results, regarding theblowing up of solution, for the NLS equation can be found in [17]. Asa classical approach to the continuous problem Brezis and Gallouet [9]consider the existence of a solution to the cubic Schrodinger equation ina bounded domain. The nonlinear Schrodinger equation has also beenstudied in higher dimensions; see, e.g., Tsutsumi and Hayashi [36].

In numerical studies of the NLS equation a complex-valued versionof Brouwer fixed point theorem is used for proving existence of the dis-cretized form of the equation, while Newton’s method has been widelyused to handle the nonlinear term numerically. In this setting Akrivis[1] propose a finite difference scheme combined with Newton’s methodfor approximation of the cubic Schrodinger equation in one dimension.A modified Newton method is introduced by Akrivis et al. in [2]. Themain idea of this modified method is to avoid updating the matrix of thelinear system at each time step. An alternative approach can be foundin [39], where a linear two-step finite element method is introduced andthe nonlinear term is approximated by the disretized solution in previoustime step.

4. The streamline diffusion method

Both Vlasov and Maxwell equations are hyperbolic type PDEs and as-suming that the exact solution is in the Sobolev space Hk+1, the standardGalerkin method for hyperbolic partial differential equations is subopti-mal in the sense that they have a convergence rate of order O(hk), whereh is the mesh size. On the other hand, with the same regularity assump-tion (Hk+1) the optimal convergence rate for the elliptic and parabolicproblems is of order O(hk+1). The streamline diffusion (SD) scheme,in comparison to the classical finite element method for hyperbolic typeequations, is more stable and has an improved convergence rate of or-der O(hk+1/2). The SD method is designed based on a Petrov-Galerkinscheme where an extra convection term in the streamline direction hasbeen added to the test functions. This corresponds to the addition ofartificial diffusion to the original equation, which enhance the stabilityand improves the convergence behavior of the standard Galerkin method

5

The streamline diffusion method

by a factor of order O(h1/2). This is the main reason for studying the SDmethod for the Vlasov-Maxwell system.

The streamline diffusion method for hyperbolic partial differentialequations has been suggested by Hughes and Brooks in [25]. For a mathe-matical analysis of the SD method; see, e.g., [4] for Vlasov-Poisson systemand Johnson and Saranen [27] for Euler and Navier-Stokes equations. Tojustify for the SD method let us look at an example in one dimensionalcase and illustrate the method. Suppose we want to approximate thesolution of the Dirichlet boundary value problem ut + ux = f(t, x) for 0 < x < 1 and t > 0,

u(t, 0) = u(t, 1) = 0 for t > 0,u(0, x) = g(x) for 0 < x < 1.

Using the classical finite element method we want to find an approxima-tion Uh in some finite element space V h, such that the following discreteversion of the variational formulation holds true

(Uht + Uhx , v) = (f, v) ∀v ∈ V h.

For the streamline diffusion method we will instead search for Uh ∈ V h,such that(

Uht + Uhx , v + δ(vt + vx))

=(f, v + δ(vt + vx)

)∀v ∈ V h.

Here δ is a multiple of h (or more general a multiple of hα for somesuitable α > 0). In the SD scheme, which is used in this thesis, we willalso allow jump discontinuties at grid points in the time direction: Let0 = t0 < t1 < . . . < tM−1 < tM = T be a partition of [0, T ] into thesubintervals Im = (tm, tm+1], m = 0, . . . ,M − 1. Then, the SD schemefor the example above can be formulated as: find Uh ∈ V h, such that form = 0, 1, . . . ,M − 1,(

Uht + Uhx , v + δ(vt + vx))Im×Ω

+

∫ΩUh+(tm, x) v+(tm, x) dx

=(f, v + δ(vt + vx)

)Im×Ω

+

∫ΩUh−(tm, x) v+(tm, x) dx ∀v ∈ V h,

where v±(t, x) = lims→0± v(t + s, x). Roughly speaking, the two extraintegrals measure how large the jumps are.

With v = Uh the extra δ term can be interpreted (after partial inte-gration) as −δ(Uhtt + Uhxx) corresponding to an extra diffusion of order δ

6

Nitsche’s method

in the streamline direction which is not present in the equation. Hencethe name of the method.

5. The hp-version

For the classical finite element method (FEM) the accuracy is usuallycontrolled by altering the mesh size h, while the polynomial degree p of theelements are kept fixed, which leads to the so-called h-version of the FEM.Alternatively, one could keep the mesh size fixed and instead increase thespectral order p. Then, this is called the p-version. Combining these twoversions leads to the hp-version. The main idea is to split the physicaldomain of the equation according to the degree of regularity of the exactsolution, data and geometry of the domain. Then, e.g., where the exactsolution is smooth one uses higher polynomial order and a coarse mesh,while near singularities a more refined mesh is applied together with alow polynomial order. In that sense the hp method may be interpretedas an automatic adaptivity procedure.

An early study of the hp-version can be found in [7], where the authorsanalyze the hp-method for an elliptic equation. In [23] and [24] Hous-ton et al. combine the hp-version with a streamline diffusion method forhyperbolic equations. A similar study can be found in [6] for the Vlasov-Poisson-Fokker-Planck system.

6. Nitsche’s method

In [32] Nitsche proposed a stabilized penalty method for an ellipticequation. This method has thereafter been widely studied by severalauthors for different equations. Arnold [3] extended Nitsche’s methodto the heat equation. In a series of articles [10], [11] and [34] the au-thors study Nitsche’s method for different equations. In [10] and [11] theNitsche method is applied to a domain decomposition problem for Pois-son and Stokes equations, respectively. Sticko and Kreiss [34] extendedthis method to the wave equation.

To illustrate Nitsche’s method we look at a simple example as follows.Suppose we are interested in approximating the solution of the Poissonequation

(6.1)

−∆u(x) = f(x) in Ω,

u(x) = α on ∂Ω,

7

Summary of papers I-III

for given function f and constant α. Here Ω is a domain in Rn for n ≥ 2.For a suitable finite dimensional space V h the classical finite elementmethod can be formulated as: find Uh ∈ V h with Uh|∂Ω = α, such that

(∇Uh,∇v) = (f, v) ∀v ∈ V h0 ,

where V h0 = v ∈ V h; v = 0 on ∂Ω. If we remove the restriction of

Uh being equal to α on the boundary and instead enforce the boundarycondition weakly (equivalently assume also v ∈ V h), then we have thefollowing formulation: find Uh ∈ V h, such that

(∇Uh,∇v)−∫∂Ω

∂Uh

∂nv ds−

∫∂Ω

(Uh − α)∂v

∂nds = (f, v) ∀v ∈ V h.

This, however, is not satisfactory enough, since the corresponding bilinearform is not gauranteed to be coercive. To compensate for the loss ofcoercivity a particular additional boundary term is added. This yieldsthe Nitsche scheme, which can be formulated as follows: find Uh ∈ V h,such that

(∇Uh,∇v)−∫∂Ω

∂Uh

∂nv ds−

∫∂Ω

(Uh − α)∂v

∂nds

+γ

h

∫∂Ω

(Uh − α)v ds = (f, v) ∀v ∈ V h.

(6.2)

Here h is the mesh size and γ is a penalty parameter to be chosen largeenough for the bilinear form to be coercive. A simple calculation showsthat the exact solution of (6.1) satisfies (6.2) meaning that Nitsche’smethod is consistent. Since the boundary condition is enforced throughthe forth term of (6.2) the third term could be removed, but then thebilinear form will not be symmetric.

7. Summary of papers I-III

Papers I and II are devoted to the analysis of the streamline diffusionscheme for the one and one-half dimensional relativistic Vlasov-Maxwellsystem in the a priori and a posteriori settings, respectively. Since thesystem is both nonlinear and coupled, we perform an iteration procedureto approximate the solution of the Vlasov-Maxwell system (2.1). Startingwith an initial guess fh,0 we compute the fields Eh,1 and Bh,1 and insertthem into the Vlasov equation to get the numerical approximation fh,1

at the next step. This will then be inserted in the Maxwell equations to

8


get the corresponding fields Eh,2 and Bh,2 and so on. The iteration stepi yields a Vlasov equation for fh,i with the fields Eh,i and Bh,i.

In Paper I we focus on the a priori error analysis of the scheme de-scribed above, while Paper II is devoted to the a posteriori error analysis.In Paper I we also consider a SD-based discontinuous Galerkin scheme,where in addition to grid points in time, jumps are also allowed acrossinter-element boundaries in phase-space variables. The a priori error es-timates are derived in two different triple norms (energy norms), one forthe Maxwell part and another for the Vlasov part. Paper III generalizethe SD scheme introduced in Paper I to a hp-scheme for the three di-mensional VM system. This paper contains also the study of a Nitschetype scheme for the Maxwell equations. As in Paper I, the focus is on apriori error estimates. A concise numerical implementation is justifyingthe results.

Before stating our results from the three VM papers, we need tointroduce some notations. Let Ωx ⊂ R and Ωv ⊂ R2 denote the spaceand velocity domains, respectively. Now we will introduce a finite elementstructure on Ωx × Ωv. Let T xh = τx and T vh = τv be finite elementssubdivisions of Ωx with elements τx and Ωv with elements τv, respectively.Then Th = T xh ×T vh = τx×τv = τ is a subdivision of Ωx×Ωv. Let 0 =t0 < t1 < . . . < tM−1 < tM = T be a partition of [0, T ] into sub-intervalsIm = (tm−1, tm], m = 1, 2, . . . ,M . Further let Ch be the correspondingsubdivision of QT = [0, T ] × Ωx × Ωv into elements K = Im × τ , with

h = diamK as the mesh parameter. We also introduce Ch as the finiteelement subdivision of QT = [0, T ] × Ωx, which is considered for theMaxwell part solely. Before we define our finite dimensional spaces weneed to introduce some function spaces, viz

H0 =

M∏m=1

H10 (Im × Ωx × Ωv) and H0 =

M∏m=1

H10 (Im × Ωx),

whereH1

0 (Im × Ω) = w ∈ H1;w = 0 on ∂Ωwith Ω = Ωx × Ωv for VM or Ω = Ωx for the Maxwell part. For k =0, 1, 2, . . ., we define the finite element spaces

Vh = w ∈ H0;w|K ∈ Pk(Im)× Pk(τx)× Pk(τv), ∀K = Im × τ ∈ Chand

Vh = g ∈ [H0]3; gi|K ∈ Pk(Im)×Pk(τx), ∀K = Im× τx ∈ Ch, i = 1, 2, 3,9


where Pk(·) is the set of polynomial of degree at most k on the given set.We shall also use some notation as

(f, g)m = (f, g)Sm =

∫Sm

f(t, x, v)g(t, x, v) dt dx dv, ‖g‖m = (g, g)1/2m

and

〈f, g〉m = (f(tm, . . .), g(tm, . . .))Ω, |g|m = 〈g, g〉1/2m ,

where Sm = Im×Ω, is the slab at m-th time interval Im, m = 1, 2, . . . ,M .Here, the domain Ω equals Ωx for the Maxwell equations, while Ω =Ωx × Ωv for the Vlasov equation.

To proceed we define fh,i, bh,i and W h,i as the approximation on theith step for f , b and W , respectively. We are now ready to formulate theSD scheme for (2.2). This is stated separately for the Maxwell equationsand the Vlasov equation. The streamline diffusion method on the ith stepfor the Maxwell part can now be formulated as follows: Find W h,i ∈ Vhsuch that for m = 1, 2, . . . ,M ,

(7.1) (M1Wh,it +M2W

h,ix , g + δ(M1gt +M2gx))m + 〈W h,i

+ , g+〉m−1

= (bh,i−1, g + δ(M1gt +M2gx))m + 〈W h,i− , g+〉m−1, ∀ g ∈ Vh,

where g = (g1, g1, g2, g3)T . The corresponding streamline diffusion methodon the ith step for the Vlasov equation can be formulated as follows: Findfh,i ∈ Vh such that for m = 1, 2, . . . ,M ,

(fh,it +G(fh,i−1) · ∇fh,i, g+δ(gt +G(fh,i−1) · ∇g))m

+ 〈fh,i+ , g+〉m−1 = 〈fh,i− , g+〉m−1, ∀g ∈ Vh,

(7.2)

where

G(fh,i−1) = (v1, Eh,i1 + v2B

h,i, Eh,i2 − v1Bh,i).

As stated above the a priori error estimates will be derived in two differenttriple norms. These triple norms are defined by

|||g|||2M =1

2

(|g+|20 + |g−|2M +

M−1∑m=1

|[g]|2m + 2δ

M∑m=1

‖M1gt +M2gx‖2m

)for the Maxwell equations and

|||g|||2V =1

2

(|g+|20 + |g−|2M +

M−1∑m=1

|[g]|2m + 2δM∑m=1

‖gt +G(fh,i−1) · ∇g‖2m

)10


for the Vlasov part.The main result of Paper I is the following convergence theorem for

the density function f .

Theorem 7.1. Let fh,i be the solution to (7.2) and assume that theexact solution f for (2.3) is in the Sobolev space Hk+1 and that someadditional conditions holds (see Paper I for details), then there exists aconstant C such that

|||f − fh,i|||V ≤ Chk+ 12 + Chi−

12 .

We have a corresponding error estimate for the solution W h,i of (7.1)and the exact solution W for (2.3). If we assume that W is in the Sobolevspace Hk+1, then there exists a constant C such that

|||W −W h,i|||M ≤ Chk+ 12 + Chi−1.

In the a posteriori setting in Paper II, the errors are measured in termsof the residuals. Due to the nature of the approximating scheme, twokinds of residuals will appear in our estimates. The first kind measureshow well the approximate solution satisfies the equation. This is donereplacing the exact solution by a symbol for an approximate solution intothe equation and taking the difference between the right and the lefthand side. The second kind of residual measures how large the jumpsof approximated solution is. Consequently, the residuals for the Maxwellequations are defined as

Ri1 = bh,i−1 −M1Wh,it −M2W

h,ix

and

Ri2|Sm =(W h,i

+ (tm, x)−W h,i− (tm, x)

)/h.

The corresponding residuals for the Vlasov equation are

Ri1 = fh,it +G(fh,i−1) · ∇fh,i

and

Ri2|Sm =(fh,i+ (tm, x, v)− fh,i− (tm, x, v)

)/h,

where the residuals with index 2 are considered to be constant in time oneach slab.

Let us decompose the error into two parts

W −W h,i = W −W i︸︷︷︸analytical iteration error

+ W i −W h,i︸︷︷︸numerical error

= E i + ei,

11


where W i is the exact solution to the approximated Maxwell equationsat the ith iteration step:

M1Wit +M2W

ix = bh,i−1.

We do a similar decomposition of the error

f − fh,i = f − f i︸︷︷︸analytical iteration error

+ f i − fh,i︸︷︷︸numerical error

= E i + ei,

where f i is the exact solution of the approximated Vlasov equation at theith iteration step:

f it +G(fh,i−1) · ∇f i = 0.

We will now assume thatW i and f i converge to the analytical solutionsWand f , respectively. For a sufficiently large i, ei and ei are the dominatingpart of the errors W −W h,i and f − fh,i. Therefore, for large enough i,we have that

‖E i‖H−1(QT ) << ‖ei‖H−1(QT ).

As the main results in Paper II we have the following a posteriorierror estimates in L∞(H−1)-norms

Theorem 7.2. There exists a constant C such that

‖W (T, ·)−W h,i(T, ·)‖H−1(Ωx) ≤ C(‖hRi1‖L2(QT ) + ‖hRi2‖L2(QT )

).

and


‖f(T, ·)− fh,i(T, ·)‖H−1(Ω) ≤

C(‖hRi1‖L2(QT )(2 + ‖G(fh,i−1)‖L∞(QT )) + ‖hRi2‖L2(QT )

).

For the Maxwell part we also have the following a posteriori estimatein the H−1(H−1)-norm


‖W −W h,i‖H−1(QT ) ≤ C(‖hRi1‖L2(QT ) + ‖hRi2‖L2(QT )

).

In Paper III the SD scheme, introduced in Paper I, is generalized toa hp-scheme for the three dimensional VM system. The a priori analysisof the hp-scheme in Paper III follows the analysis in Paper I, with somemodification because of the use of a hp method. An important ingredientis the interpolation error estimates in Theorem 3.2 of [6]. The following

12


bounds hold for the interpolation error η = g − g of a function g ∈Hk+1([0, T ]×Ω) (where Ω stands for Ωx in the Maxwell’s equations andΩx × Ωv for the Vlasov case) and its gradient:

‖η‖2 ≤ C∑K

(hK2

)2sK+2

Φ1(pK , sK)‖g‖2sK+1,K ,

‖Dη‖2 ≤ C∑K

(hK2

)2sK

Φ2(pK , sK)‖g‖2sK+1,K ,

where the sums are taken over all space-time elements of the triangulationof the domain, [0, T ] × Ω, 0 ≤ sK ≤ min(pK , k), with pK being thespectral order of the element K. The norm ‖ · ‖s,K is the standard normon Hs(K). Closed form formulas for Φ1 and Φ2 are given in Theorem 3.2of [6]. A less involved formula for Φ1 can be found in [24]. In the errorestimates we use ΦM = max(Φ1,Φ2) with dimension N = dim Ωx+1 andΦV = max(Φ1,Φ2) with dimension N = dim Ωx + dim Ωv + 1. The maintheorems of the hp-scheme for the Vlasov-Maxwell system consist of theerror estimates

|||W −W h,i|||2M ≤C∑K∈Ch

h2sK+1K p−1

K ΦM(pK , sK)‖W‖2sK+1,K

+ C‖f − fh,i−1‖2QT,

for the fields and

|||f − fh,i|||2V ≤ C( ∑K∈Ch

h2sK+1K p−1

K ΦM(pK , sK)‖W‖2sK+1,K

+∑K∈Ch

h2sK+1K p−1

K ΦV(pK , sK)‖f‖2sK+1,K

)+ Cph‖f − fh,i−1‖2QT

for the density function. Note that the triple norms appearing here arenot exactly the same as in Paper I. The explicit expressions of the triplenorms and the relevant conditions for the above error estimates can befound in Paper III. The results are justified through numerical simulationsin a lower dimensional case.

In Paper III a Nitsche scheme is studied for the Maxwell equations aswell. This can be viewed as an alternative method to the SD scheme for

13


the Maxwell part. By assuming extra regularity and differentiating theMaxwell’s equations

Et −∇x ×B = −j,Bt +∇x × E = 0,

with respect to t, we obtain second order differential equations

Ett +∇x × (∇x × E) = −jt,Btt +∇x × (∇x ×B) = ∇x × j.

Since the equations for E and B are similar, it suffices to analyze theequation for one of them: E. To this end we define the bilinear form

a(E, g) :=

∫Ωx

∇x × E · ∇x × g dx−∫

Γx

∇x × E · (g × n) ds

−∫

Γx

(E × n) · ∇x × g ds+γ

h

∫Γx

E · g ds

and the finite element space

V xh = g ∈ H(curl,Ωx) : g|τx ∈ P1(τx), ∀τx ∈ T xh .

The semi-discrete problem can now be formulated as: for each fixed t,find Eh(t, ·) ∈ V x

h , such that

(Ehtt, g)Ωx + a(Eh, g) = −(jt, g)Ωx ∀g ∈ V xh .

The continuity and coercivity for the bilinear form a(·, ·) hold in the meshdependent norm

|||g|||2h := ‖∇x × g‖2Ωx+ ‖h−1/2g‖2Γx

+ ‖h1/2∇x × g‖2Γx.

However, the coercivity is only valid when the penalty parameter γ ischosen large enough (see Paper III for more details on how to choose γ).An important feature in the analysis of the Nitsche method is the use ofthe projection Qh : H(curl,Ωx)→ V x

h defined by

a(Qhu, v) = a(u, v) ∀v ∈ V xh ,

with the following error estimates

Lemma 7.5. There exists a constant C, such that

‖u−Qhu‖Ωx + h|||u−Qhu|||h ≤ Ch2‖u‖H2(Ωx).

14

Summary of paper IV

To formulate the fully discrete problem we define the notations

∂2t u

m =um − 2um−1 + um−2

k2,

um =um + 2um−1 + um−2

4,

where um = u(tm). Then, the fully discrete problem for the Maxwellpart, reads as follows: for m = 2, 3, . . . ,M , find Em such that

(∂2t Em, χ) + a(Em, χ) = −(jm−1

t , χ) ∀χ ∈ V xh .

Here E0 = QhE0 and E1 is assumed to be chosen as an approximation

of E1 satisfying ‖E1 − E1‖Ωx ≤ C(h2 + k2). Then we have the followingerror estimate for the fully discrete problem:

‖E(tm)− Em‖Ωx ≤ C(h+ k).

8. Summary of paper IV

Paper IV is devoted to a two level time stepping scheme for the non-linear Schrodinger equation (3.2). To formulate the discretization pro-cedure, we introduce a not necessarily uniform partition tnNn=0 of thetime interval [0, T ], i.e. tn < tn+1 for n = 0, 1, . . . , N − 1 , t0 = 0 andtN = T . Then, we set kn := tn − tn−1 for n = 0, 1, . . . , N . For a suitablefinite element space Sh, the following two time-step numerical scheme isanalyzed in Paper IV:

Step 1. Set

U0h = u0,h,

where u0,h = Rhu0 is the Ritz projection of the initial data.

Step 2. For n = 1, 2, . . . , N , first find Un− 1

2h ∈ Sh such thatUn− 1

2h − Un−1

h

kn/2, χ

=i

∇Un− 12

h +∇Un−1h

2,∇χ

+ i

f(|Un−1n |2)

Un− 1

2h + Un−1

h

2, χ

+ (g(tn−1, ·), χ) ,

15

Summary of paper IV

for all χ ∈ Sh and then find Unh ∈ Sh such that(Unh − U

n−1h

kn, χ

)=i

(∇Unh +∇Un−1

h

2,∇χ

)

+ i

(f(|Un−

12

h |2)Unh + Un−1

h

2, χ

)+(g(tn− 1

2, ·), χ

),

for all χ ∈ Sh.An important step in the analysis of the above scheme is the consis-

tency analysis. For n = 1, 2, . . . , N , we define rn−12 and rn by

un−12 − un−1

kn/2=i∆

(un− 12 + un−1

2

)+ if(|un−1|2)

un−12 + un−1

2

+ g(tn−1, ·) + rn−12

and

un − un−1

kn=i∆

(un + un−1

2

)+ if(|un−

12 |2)

un + un−1

2

+ g(tn− 12, ·) + rn,

respectively, where un = u(tn, ·) for n = 0, 1, . . . , N . The terms rn−12 and

rn measure how consistent our time discretization is with the original

equation. We have the following estimates for rn−12 and rn:

Proposition 8.1. Assume that there is a constant C1 such that

(8.1) max(‖∂tu‖∞, ‖∂2

t u‖∞, ‖∆∂tu‖∞)< C1,

then

‖rn−12 ‖ ≤ Ckn.

Further if, in addition to (8.1), we have that there is a constant C2 suchthat

(8.2) max(‖∂3

t u‖∞, ‖∆∂2t u‖∞

)< C2,

then

‖rn‖ ≤ Ck2n.

The main result of Paper IV is given by the following theorem.

16

References

Theorem 8.2. Let en := Unh −un be the error at the time level t = tn.Assume that u satisfies the conditions (8.1) and (8.2). Then there is aconstant C such that

‖en‖ ≤ C(k2 + hr),

and

‖∇en‖ ≤ C(k + hr−1),

with k := max1≤n≤N kn.

One would expect an order of O(k2 + hr−1) for the error estimate ofthe gradient, since we only have spatial derivatives (cf. [39]).

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