# Convergence Analysis of a Fourier-Based Solution Method

Authors: John L. Fleming
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

When engineers model the magnetostatic fields applied to recording
heads of computer hard drives due to a magnetic recording medium,
the solution of Laplace's equation must be found. A popular solution
method is based on Fourier analysis. However, due to the geometry of
the read head model, an interesting mathematical problem arises. The
coefficients for the Fourier series solution of the desired magnetic
potential satisfy an infinite system of linear equations. In
practice, the infinite system is truncated to a finite system with
little consideration for the effect this truncation has on the
solution. The paper will provide a proper understanding of the
underlying problem and a formal analysis of the effect of
truncation.


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## Body

## 1. Introduction

Engineering models of magnetic recording seek to simulate the
magnetic potential that a shielded magnetoresistive head will experience as it
passes over a magnetized recording medium. Finding the magnetic potential
requires the solution of the Laplace equation. Often, a simple rectangular-type
geometry is employed which naturally leads to the use of Fourier-based
solutions. The desired potential can be found formally by some fairly
straightforward calculations. However, in spite of the apparent simplicity,
there is an important mathematical issue which arises in these calculations.
Since the models involve a transition from an empty half-space to slot with a
finite width, the Fourier coefficients of the solution satisfy an infinite
system of linear equations. In order to compute a solution, the infinite system
must be truncated to a finite system. From a mathematical and numerical
analysis point of view, the effect of this truncation is not well understood or
adequately explained. The main goal of this  paper is to provide the proper
mathematical context under which the effect of the truncation can be
characterized.


## 1.1. Problem Description

Consider
Laplace's equationΔϕ=0 in a
two-dimensional setting. The solution ϕ represents a
magnetic scalar potential. Further, assume that the xy-plane is
divided into two half-spaces. In the lower half-space (y<0), there is an
empty space containing a charge which will induce a potential. In the upper half-space
(y>0), assume that
there is a perfect magnetic conductor with a gap of empty space from x=0 to x=G. The empty space in region I will be referred to as
“the gap,” while the conducting material in region I will be called
“the shields.” Note that the shields are assumed to extend to
infinity in both the x- and y-directions.
The geometry indicates that the potential is only nonzero in the lower half-space
and in the gap between the shields in the upper half-space (see Figure 1).
Also, note that the solution will satisfy a Dirichlet boundary condition (ϕ=0) at any
interface with the shields.Figure 1
Diagram of solutionS domain for the variational problem.A special boundary condition will be provided at the
interface between region I and region II (y=0). The boundary
condition will ensure continuity of the potential and its normal derivative
across the interface and also will properly reflect the influence of the charge
in region II on the potential in region I. In addition, an artificial boundary
will be placed in the gap at some positive value y=b inside the gap
in region I. The artificial boundary will create a finite rectangular domain
which will be used later in the paper for the analysis of the variational form
of the problem. The region within the gap, above the interface and below the
artificial boundary, will be denoted as S=[0,G]×[0,b]. The boundary of the region S is denoted as Γ. In particular, the portion of the boundary at y=b is denoted as Γ1, while the portion of the boundary at y=0 is denoted as Γ2.The geometry described here has been used as a
simplified model of the read process in magnetic recording [1–4]. A very
similar situation using the Helmholtz equation is used to describe the
scattering of plane waves from a groove in a perfect electric conductor [5, 6].
Another similar situation occurs when dealing with waveguide junctions [7, 8].
An approach of basic separation of variables allows us to find the form of the
solutions in the lower and upper half-spaces in terms of their Fourier
components. The aim is to find the solution of the Laplace equation in the gap
(region I) of the upper half-space due to a charge which exists outside the gap
in the lower half-space. However, to completely describe the solution in the
gap of the upper half-space, it is necessary to study closely what occurs at
the interface between the lower half-space and the gap (y=0).


## 1.2. Focus of the Paper

The paper will
show how to produce the Fourier solution of the potentialϕ in the gap
region described above. We will see that the solution method produces an
infinite linear system of equations whose solution represents the Fourier
coefficients of the potential in region I. The solution to the infinite system
is approximated by solving a finite truncated version of the linear system.
From a mathematical point of view, the truncation needs to be validated. In
principle, there is no guarantee that the finite version even has a unique
solution. Also, given the existence of the solution, it must be confirmed that
the finite problem converges to the actual solution as the size of the finite
system increases. Finally, if the problem does converge, we would like to
characterize the rate at which the finite Fourier series converges to the
actual series solution.As we will see below, the nature of the infinite
system is not easily conducive to mathematical or numerical analysis. As a
novel approach, the paper will consider the variational solution of the problem
which will provide an alternative avenue for analysis. It will be shown that in
the proper context, the variational formulation is equivalent to the truncated
finite version of the Fourier approach. Once this equivalence is established,
the power of functional analysis will allow for a fairly straightforward
analysis of the Fourier-based solution in the context of the variational
solution.The intent of the paper is not only to analyze the
convergence of the finite approximations of the infinite linear system, but
also to provide a perspective on the problem which is accessible to those
familiar with functional analysis. The same theory which describes the
convergence of finite element approximations will ultimately be applied to the
finite Fourier-based approximations. Another benefit of this unique perspective
is that it will allow for the description of the convergence properties without
any need for direct analysis of the infinite linear system itself. One will
simply need to determine how well the finite Fourier expansions are able to
approximate the functions which are in the solution space of the variational
problem.


## 2. Fourier Solution

The
Fourier-based solution method for the geometry in Figure1 was offered in [2, 3]. The method presented, or very similar methods, has been used within
several models [1, 5, 6]. These papers offer a pragmatic method for finding the
solution with Fourier theory, but they offer no mathematical justification of
some important details of the method. In particular, as will be shown below,
the solution of the Fourier methods satisfies an infinite linear system of
equations. Approximate solutions are found by truncating these systems, but the
mathematical analysis of this approach is lacking. Some analysis of a similar
problem involving transitions in waveguides has been done where the infinite
system is analyzed directly [8]. Some analysis of infinite matrices arising
from solutions to Poisson's equation has been done [9, 10]. Also, analysis of
infinite matrices using a matric operator technique for scattering problems has
been done [11]. None of the mentioned works considers the alternative approach
for the analysis using the variational form of the problem.The Fourier solution is found by studying what happens
at the transition from region I to region II. In region I, the solution will
have the following form:(2.1)ϕI=∑n=1∞Ansin(nπxG)e−(nπ/G)y.  Since the
support of the potential in the x-direction is
the interval from 0 to G, a Fourier sine series can be used. On the other
hand, in region II the solution has the form (2.2)ϕII=∫−∞+∞(B(kx)e−κy+C(kx)eκy)e2πikxxdkx,  where κ=|2πkx|. Since the support in the x-direction is (−∞,∞) in region II, a
Fourier transform must be used. The interesting part of the problem comes from
coupling the two forms of the solution together at the gap interface.Recall that the main problem is to compute the
potential in region I given a charge distributionρ in region II.
In terms of the Fourier approach, we must compute the Fourier coefficients An in (2.1).


## 2.1. Finite Fourier Approximate Solution Method

The finite
Fourier method to compute the Fourier coefficients has been presented in
multiple papers [2, 3]. The solution comes from an examination of what happens
at the interface between the gap in region I and the lower half-plane of region
II. The two forms of the solution will satisfy some standard continuity conditions.First, the potential is continuous at the gap
interface. This fact gives the following equation:(2.3)ϕI|y=0=ϕII|y=0.  Second, the
normal derivative (∂/∂y) is continuous
across the gap interface. Hence, we have (2.4)∂ϕI∂y|y=0=∂ϕII∂y|y=0.  The An's are computed
by applying the continuity conditions to the two forms of the solution (2.1)
and (2.2).The first continuity equation gives the following:(2.5)∑n=1∞Ansin(nπxG)=∫−∞+∞(B+C)e2πikxxdkx  or (2.6)∑n=1∞Ansin(nπxG)=ℱx−1(B+C).  Taking the
Fourier transform in the x-direction of
both sides gives (2.7)∑n=1∞Anℱx(sin(nπxG))=(B+C)  or (2.8)ℱx(ϕI)=B+C.  Note that both
the An's and C are unknown.
The B can be computed
directly using a Green function and the known charge distribution ρ in region II.
Hence, it is necessary to eliminate C which allows
for the determination of the An's in terms of
the known quantity B. The second continuity condition allows for the
elimination of C from the
problem.The second continuity condition gives the following:(2.9)∑n=1∞(−nπG)Ansin(nπxG)=∫−∞+∞(−κB+κC)e2πikxxdkx  or (2.10)∑n=1∞(−nπG)Ansin(nπxG)=ℱx−1(−κB+κC).  Multiplying  both sides of the previous equation by sin(mπx/G) and integrating
them from 0 to G give the
following: (2.11)−(2G)(mπG)Am=∫0Gℱx−1(−κB+κC)sin(mπxG)dx.  Solve for C in (2.8) and
substitute it into (2.11) to arrive at the
following: (2.12)−2mπG2Am=∫0Gℱx−1[−2κB+∑n=1∞Anℱx(sin(nπxG))]sin(mπxG)dx.  By Parseval's
formula, we have (2.13)−2mπG2Am=−Bm+∑n=1∞KmnAn,  where (2.14)Bm=∫−∞+∞2Bℱx−1(sin(mπxG))dkx,Knm=∫−∞+∞κℱx(sin(nπxG))ℱx−1(sin(mπxG))dkx.Thus, we have a formulation of the Fourier
coefficients in terms of the Fourier transformB. However, we see that the Fourier coefficients of ϕI satisfy an
infinite system of linear equations given by (2.13). Denote the infinite system
as (2.15)K˜{A¯}={B¯},  where K˜nm=Knm+(−2mπ/G2)δnm.


## 2.2. The Issue of Truncation

For purposes of
computation, such an infinite system does little good. An obvious response to
deal with the situation is to solve a truncated version of the infinite matrix(2.16)KN˜{AN¯}={BN¯}.  Even though
this seems to be the logical approach to the infinite matrix issue from a
computational perspective, the truncation leads to some important mathematical
questions.(1)
Is the
truncated matrix invertible?(2)
If the truncated matrix is invertible, does the
finite version converge to the actual solution?(3)
If the finite version of the matrix converges to
the actual solution, what is the convergence rate?These very important questions must be answered to
have some reasonable level of trust in the solution method. The problem is how
to answer such questions about the truncation of the infinite matrix. Direct
analysis of the infinite system is one approach, but an alternative provided
below is to study the variational form of the problem and compare it to the
Fourier method.


## 3. Variational Formulation

The variational
approach not only provides a basis for mathematical proofs of existence and
uniqueness, but also provides a basis for robust numerical methods including
the finite element method. First, the variational form of the problem will be
derived. Second, the variational form will be shown to have a unique solution.
Finally, it will be shown that if the variational form is applied to a
particular space of functions, then the result is equivalent to the truncated
Fourier method. Once this equivalence is established, the analysis of the
truncated Fourier method can be performed by analyzing the equivalent
variational form.


## 3.1. Derivation of the Variational Form

For the
variational formulation, we will use the finite solution domain. Look for a
solution of∇u=0 in the
rectangular region S=[0,G]×[0,b]. We will use u to represent
the solution of the variational problem. Ultimately, the solution will be
equivalent to the potential ϕI found by the
Fourier approach. Recall that Γ will denote the
boundary of S. To enforce the boundary conditions u(0,y)=u(G,y)=0, we will look for a solution in the space of
functions (3.1)H˜01(S)={w∈H1(S)∣w(0,y)=w(G,y)=0}.  The weak form
of the problem is constructed as follows.(1) Multiply
both sides of the equation∇u by a function w in H˜01(S) and integrate
them over S: (3.2)∫SΔuwdS=0.(2) Apply
integration by parts:(3.3)−∫S∇u·∇wdS+∫Γ∂u∂nwdΓ=0.  Here, we have
the weak form or the variational form of the problem. We must examine the
boundary integral term of the weak form of the equation. The boundary
conditions must be constructed for the interface between region I and region II
as well as for truncating region I domain to be finite.We know thatw(0,y)=w(G,y)=0, but we must also specify boundary conditions on Γ1(y=b) and Γ2(y=0). First, on Γ1 we know that
the solution will be of the form (3.4)u=∑n=1∞Ansin(nπxG)e−(nπ/G)y.  Therefore, on Γ1 we have (3.5)∂u∂n=∂u∂y=∑n=1∞(−nπG)Ansin(nπxG)e−(nπ/G)b.  Hence, a
Dirichlet to Neumann map T1(u) can be defined
on Γ1 by (3.6)∂u∂n=T1(u)=∑n=1∞(−nπG)Ansin(nπxG)e−(nπ/G)b.On the boundaryΓ2, we must refer back to the second matching condition: (3.7)∂u∂y=ℱx−1(−κB+κC).  Again, solve
(2.7) for C and substitute it into the previous
equation to get (3.8)∂u∂y=ℱx−1(−2κB+κℱx(u))  or (3.9)∂u∂y=ℱx−1(κℱx(u))+g,  where g=−2ℱx−1(κB).Hence, a Dirichlet to Neumann mapT2(u) can be defined
on Γ2 as follows: (3.10)∂u∂n=T2(u)+g=−ℱx−1(κℱx(u))−g.Now, put the mapsT1(u) and T2(u) into the weak
form of (3.3): (3.11)−∫S∇u·∇wdS+∫Γ1T1(u)wdΓ+∫Γ2T2(u)wdΓ=∫Γ1gwdΓ.  The weak form
of the equation defines a bilinear form (3.12)a(u,w)=−∫S∇u·∇wdS+∫Γ1T1(u)wdΓ+∫Γ2T2(u)wdΓ  and a bounded
linear functional (3.13)〈g,w〉=∫Γ1gwdΓ.  The solution of
the variational form of the problem is a function u∈H˜01(S) such that (3.14)a(u,w)=〈g,w〉  for all w∈H˜01(S).


## 3.2. Existence and Uniqueness

The weak or
variational formulation is very useful for proving existence and uniqueness of
the problem at hand. The established theory for elliptic differential equations
can be used here as in the following theorem.Theorem 3.1.
The
variational problem (3.14) has a unique solution in  H˜01(S).The
existence/uniqueness proof can be found in [12]. The proof establishes the
continuity and coercivity of the variational form in order to apply the
Lax-Milgram theorem to verify existence and uniqueness.The unique solutionu of the
variational problem represents the potential ϕI in region I.
The finite Fourier approach produces an approximation of ϕI. At this point, the formal variational solution
result can be used to analyze the approximate Fourier solution.


## 4. The Finite Fourier Method Versus the Variational Form

## 4.1. Existence of the Finite Fourier Solution

The connection between the finite Fourier series
method and the variational formulation can be found by considering subspaces ofH˜01(S). If a space VN is a subspace
of H˜01(S), then there is a unique solution to the variational
problem over the subspace as well, because continuity and coercivity apply on
the subspace in the same way as on the entire space [13]. Using this idea and
Theorem 3.1, we want to compare the Fourier solution method and the variational
form. The first conclusion which can be drawn from the comparison is that the
truncated matrix from the Fourier method is invertible. We will see that this
fact is just a simple corollary of the uniqueness result for the variational
problem.Define a finite subspace ofH˜01(S): (4.1)VN={uN∈H˜01(S)∣uN=∑n=1NAnsin(nπxG)e(−nπ/G)y}.  There is a
unique solution uN to the
variational problem in this space which satisfies (4.2)a(uN,v)=〈g,v〉  for all
functions v in the space VN. If we examine the variational form of the problem
when the functions come from the subspace VN, the connection can be made to the finite Fourier
solution.Solve the variational form of the problem inVN using the
Galerkin method. Let vm=sin(mπx/G)e(−mπ/G)y and carry out
the following calculations: (4.3)−∫S∇uN⋅∇vmdS=∫0b2G(mπG)Ame(−mπ/G)ydy=[(mπG)(2G)Ame(−2mπ/G)b−(mπG)(2G)Am],∫Γ1T1(uN)vmdΓ=∫0G[∑n=1N−(nπG)Ansin(nπxG)·e(−nπ/G)b]sin(mπxG)e(−mπ/G)bdx=−(mπG)(2G)Ame(−2mπ/G)b,∫Γ2T2(uN)vmdΓ=−∫0Gℱx−1(κℱx(uN))sin(mπxG)dx=∫−∞∞κℱx(uN)ℱx−1(sin(mπxG))dkx.Continuing with the last integral,(4.4)∫−∞∞ℱx(uN)ℱx−1(sin(mπxG))dkx=∑n=1N∫−∞∞Anℱx(sin(nπxG))ℱx−1(sin(mπxG))dkx.Notice that this last expression is exactly(4.5)−∑n=1NKnmAn.  Putting all the
integrations together, we get (4.6)a(uN,vm)=−(mπG)(2G)Am−∑n=1NKnmAn.Also, note that(4.7)〈g,vm〉=−∫−∞∞2Bℱx−1(sin(mπxG))dkx  which is
exactly −Bm from the
Fourier method. Therefore, we see now that by solving the variational problem
with the Galerkin method, (4.8)a(uN,vm)=〈g,vm〉  produces the
exact same matrix as the truncated matrix from the Fourier method. (4.9)KN˜{AN¯}={BN¯}.  Therefore, we
have the desired equivalence between the finite Fourier method and the variational
form of the problem on the subspace VN. In particular, the solution AN¯ of (4.9)
is the coefficients of uN. Hence, we can answer the first question about the
Fourier method. The truncated matrix is invertible. Since the variational form
has a unique solution in VN, we must conclude that the finite system representing
the solution must be nonsingular.


## 4.2. Convergence of Fourier Method Solution

We can use analysis of the variational formulation on
the subspaceVN to determine
the convergence properties of the Fourier series method.Theorem 4.1.
The
finite Fourier method converges to the true solution of the problem inH˜01(S).Proof.
Cea's
lemma   [13] states that the solution uN of the
variational problem in VN is the best
possible approximation of the true solution u in H˜01(S): (4.10)∥u−uN∥≤C(infu˜∈VN∥u−u˜∥).  By letting N→∞, we see that the finite Fourier method will converge
to the actual solution.Also, Cea's lemma allows us to answer the third
question about the Fourier method regarding the convergence rate. Cea's lemma
indicates that the solutionuN from the
subspace VN is the best
possible approximation of the true solution u in H˜01(S). Therefore, we need to determine how
“close” functions in VN can be to
functions in H˜01(S) with respect to
their norm.It can be shown that the approximationsuN will converge
to the actual series solution (4.11)u=∑n=1∞Ansin(nπxG)e(−nπ/G)y  with error of
order N−1 in H˜01(S). The key to verifying this is to show that the trace
of the first derivatives of u is actually integrable
on the boundary Γ2. This can be done since the solution of the
variational form of the problem can be shown to exist in the space H3/2(S) [14]. Therefore,
the solution has enough regularity to conclude that ∥u−un∥≤C/N [15].


## 5. Conclusion

At transitions
of rectangular regions, Fourier-based solutions of differential equations will
produce an infinite system of linear equations. In this paper, the
Fourier-based computation of a Laplace equation inside a gap between two
parallel perfect conductors is considered. In particular, the Fourier
coefficients of the solution satisfy an infinite system of linear equations.
Approximation techniques truncate this system without mathematical
justification. The paper offered a method of analysis of the Fourier method
using the variational form of the problem. We have seen that this approach
proves that the truncation of the infinite linear system will always have a
solution and the truncation will converge as the number of unknowns is
increased.



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