On the first eigenfunction of the Dirichlet-Laplacian on the regular polygon

Suppose that is the regular polygon inscribed in the unit circle with one vertex at . Consider the Dirichlet-Laplace eigenvalue problem

This problem has solutions for a sequence of eigenvalues

Denote by the eigenfunction associated to , normalized such that . The case of the first eigenvalue is particular. Since the first eigenvalue is simple, there exists a unique normalized eigenfunction such that . This implies that the function has all the symmetries of the regular polygon. Below you can see an image of the eigenfunction in the case of the regular pentagon.

Consider now the triangles given by vertices for , where . These triangles decompose into equal slices.

Furthermore, let . Then we have, by the symmetry of the eigenfunction, that .

In the following, due to the symmetry we can infer that is an even function with respect to and is odd. The fact that the gradients undergo a rotation when transferred from to we have the matrix equality

which implies that .

Up until now we have two relations between . Any supplementary independent relation would completely determine these quantities. 

Question: Can you find such a relation?

FreeFem++ Tutorial – Part 2

Click here for the first part. Some other posts related to FreeFem: link 1, link 2.

Here are a few tricks, once you know the basics of FreeFem. If your plan is straightforward: define the domain, build the mesh, define the problem, solve the problem, plot the result… then things are rather easy. If you want to do stuff in a loop, like an optimization problem, things may get more complicated, but FreeFem still has lots of tricks up its sleeves. I’ll go through some of them.

1. Defining the geometry of the domain using border might be tricky at the beginning. Keep in mind that the domain should be on the left side of the curves definining the boundaries. This could be acheived in the parametrization chosen or you could reverse the parametrization of a particular part of the domain in the following way. Suppose you have the pieces of boundary C1,C2,C3,C4. You wish to build a mesh with the command mesh Th = buildmesh(C1(100)+C2(100)+C3(100)+C4(100));but you get an error concerning a bad sens on one of the boundaries. If you identify, for example, that you need to change the orientation of C3, you can acheive this by changing the sign of the integer defining the number of points on that part of the boundary: mesh Th = buildmesh(C1(100)+C2(100)+C3(-100)+C4(100));You could make sure that you have the right orientations and good connectivities for all the boundaries by running a plot command before trying mesh: something likeplot(C1(100)+C2(100)+C3(-100)+C4(100));should produce a graph of all your boundaries with arrows showing the orientations. This is good as a debug tool when you don’t know where the error comes from when you define the domain.
2. Keep in mind that there are also other ways to build a mesh, like square, which meshes a rectangular domain with quite a few options and trunc which truncates or modifies a mesh following some criteria. Take a look in the documentation to see all the options.
3. Let’s say that you need to build a complex boundary, but with many components with similar properties. Two examples come to mind: polygons and domains with many circular holes. In order to do this keep in mind that it is possible to define a some kind of “vectorial boundary”. Let’s say that you want to mesh a polygon and you have the coordinates stored in the arrays xs,ys. Furthermore, you have another array of integers ind which point to the index of the next vertex. Then the boundary of the polygon could be defined with the following syntax:
border poly(t=0,1; i){
x=(1-t)*xx[i]+t*xx[ind(i)];
y=(1-t)*yy[i]+t*yy[ind(i)];
label=i;
}
Now the mesh could be constructed using a vector of integers NC containing the number of desired points on each of the sides of the polygon:
   
mesh Th = buildmesh (poly(NC));

4. You can change a 2D mesh using the command adaptmesh. There are various options and you’ll need to search in the documentation for a complete list. I use it in order to improve a mesh build with buildmesh. An example of command which gives good results in some of my codes is:
   
Th = adaptmesh(Th,0.02,IsMetric=1,nbvx=30000);

   The parameters are related to the size of the triangles, the geometric properties of the griangles and the maximal number of vertices you want in your mesh. Experimenting a bit might give you a better idea of how this command works in practice.
5. There are two ways of defining the problems in FreeFem. One is with solve and the other one is with problem. If you use solve then FreeFem solves the problem where it is defined. If you use problem then FreeFem remembers the problem as a variable and will solve it whenever you call this variable. This is useful when solving the same problem multiple times. Note that you can modify the coefficients of the PDE and FreeFem will build the new problem with the updated coefficients.
6. In a following post I’ll talk about simplifying your code using macros and functions. What is good to keep in mind is that macros are verbatim code replacements, which are quite useful when dealing with complex formulas in your problem definition or elsewhere. Functions allow you to run a part of the code with various parameters (just like functions in other languages like Matlab).

I’ll finish with a code which computes the eigenvalue of the Laplace operator on a square domain with multiple holes. Try and figure out what the commands and parameters do.

int N=5; //number of holes
int k=1; //number of eigenvalue

int nb = 100;  // parameter for mesh size
verbosity = 10; // parameter for the infos FreeFem gives back

real delta = 0.1;
int nbd = floor(1.0*nb/N*delta*2*pi);

int bsquare = 0;
int bdisk   = 1;

// vertices of the squares
real[int] xs(N^2),ys(N^2);

real[int] initx = 0:(N^2-1);
for(int i=0;i<N^2;i++){
  xs[i] = floor(i/N)+0.5;
  ys[i] = (i%N)+0.5;
}


int[int] Nd(N^2);

Nd = 1;
Nd = -nbd*Nd;


// sides of the square
border Cd(t=0,N){x = t;y=0;label=bsquare;}
border Cr(t=0,N){x = N;y=t;label=bsquare;}
border Cu(t=0,N){x = N-t;y=N;label=bsquare;}
border Cl(t=0,N){x = 0;y=N-t;label=bsquare;}

border disks(t=0,2*pi; i){
    x=xs[i]+delta*cos(t);
    y=ys[i]+delta*sin(t);
    label=bdisk;
}

plot(Cd(nb)+Cr(nb)+Cu(nb)+Cl(nb)+disks(Nd));

mesh Th = buildmesh(Cd(nb)+Cr(nb)+Cu(nb)+Cl(nb)+disks(Nd));

plot(Th);

int[int] bc = [0,1]; // Dirichlet boundary conditions

load "Element_P3";
fespace Vh(Th,P2);
// variables on the mesh
Vh u1,u2;

// Define the problem in weak form
varf a(u1,u2) = int2d(Th)
(dx(u1)*dx(u2) + dy(u1)*dy(u2))+on(1,u1=0)//on(C1,C2,C3,C4,u1=1)
                               +on(bc,u1=0);
varf b([u1],[u2]) = int2d(Th)(  u1*u2 ) ;
// define matrices for the eigenvalue problem
matrix A= a(Vh,Vh,solver=Crout,factorize=1);
matrix B= b(Vh,Vh,solver=CG,eps=1e-20); 

// we are interested only in the first eigenvalue
int eigCount = k;
real[int] ev(eigCount); // Holds eigenvalues
Vh[int] eV(eigCount);   // holds eigenfunctions
// Solve Ax=lBx
int numEigs = EigenValue(A,B,sym=true,sigma=0,value=ev,vector=eV);

plot(eV[k-1],fill=1,nbiso=50,value=1);

Here is the mesh and the solution given by the above program:


			

			
											

			

FreeFem++ Tutorial – Part 1

First of all, FreeFem is a numerical computing software which allows a fast and automatized treatment of a variety of problems related to partial differential equations. Its name, FreeFem, speaks for itself: it is free and it uses the finite element method. Here are a few reasons for which you may choose to use FreeFem for a certain task:

1. It allows the user to easily define 2D (and 3D) geometries and it does all the work regarding the construction of meshes on these domains.
2. The problems you want to solve can be easily written in the program once we know their weak forms.
3. Once we have variables defined on meshes or solutions to some PDE, we can easily compute all sorts of quantities like integral energies, etc.

Before showing a first example, you need to install FreeFem. If you are not familiar with command line work or you just want to get to work, like me, you can install the visual version of FreeFem which is available here. Of course, you can find example programs in the FreeFem manual or by making a brief search on the internet.

I’ll present some basic stuff, which will allow us in the end to solve the Laplace equation in a circular domain. Once we have the structure of the program, it is possible to change the shape of the domain in no time.

Simple triangle mesh – Matlab code

There are cases when you need to define a simple, regular mesh on a nice set, like a triangle, or a polygon. There are software which are capable of creating such meshes, but most of them will require some computation time (Delaunay triangulation, plus optimization, edge swapping, etc.). I propose to you a simple algorithm for defining a mesh on a triangle. Of course, this can be extended to polygons or other cases (I used a variant of it to create a mesh of a spherical surface).

The idea is very simple: start with three points , and define the (only) initial triangle as . Then, at each step, take a triangle from the list , add the midpoints of to the list of points and replace the triangle with the four smaller triangles determined by the vertices and midpoints of . This is a basic mesh refining strategy. Thus, in short, we start with a triangle and do a number of mesh refinements, until we reach the desired side-length. Of course, at each step we should be careful not to add the midpoint of an edge two times (once for every neighbouring triangle), but in what follows, I will always remove duplicates by bruteforce in the end (not the most economic solution).

Regularity of the weak solution Part 1

I will present here how to recover the regularity of a weak solution for the Dirichlet problem. The arguments can easily be adapted to most of the weak formulations involving the Laplace operator, the essential tool being the estimate of the norm of the derivatives of the solution . The arguments are adapted from H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Chapter 9, and the tool named the method of translations is due to L. Niremberg.

Consider the problem

whose weak variational form is

Note that in the variational form we only assume , and to be able to recover the PDE we need to use Green’s formula, which is valid if .

Weak formulation for Laplace Equation with Robin boundary conditions

Consider an open set with Lipschitz boundary and consider on the following problem

where is a constant. This is the Laplace equation with Robin boundary conditions. I will prove that the problem is well posed and for each there exists a solution .

Finite Difference Method for 2D Laplace equation

[Edit: This is, in fact Poisson’s equation.]

[For solving this equation on an arbitrary region using the finite difference method, take a look at this post.]

I will present here how to solve the Laplace equation using finite differences 2-dimensional case:

Lax Milgram application

Let and . Consider the bilinear form

1. Check that is a continuous symmetric bilinear form and that implies .

2. Prove that is coercive.

3. Deduce that for every there exists a unique satisfying

What is the corresponding minimization problem?

4. Show that the solution of belongs to (and in particular ). Determine the equation and the boundary conditions satisfied by .

5. Assume that , and let be the solution of . Prove that belongs to for every . Show that if and only if .

6. Determine explicitly the solution of when is a constant.

7. Set , where is the solution of and . Check that is a self-adjoint compact operator from into itself.

8. Study the eigenvalues of .

H. Brezis, Functional Analysis

One dimensional trace of a Sobolev function

Check that the mapping from to is a continuous linear functional on . Deduce that there exists a unique such that

Show that is the solution of some differential equation with appropriate boundary conditions and compute explicitly.

H. Brezis, Functional Analysis, Ex 8.18

Cinetic Equations Problem

Cinetic Models Application

Consider the following BGK type model

 

where

and

I – Let be the solution of with .

1. Prove that .

2. Prove that .

3. Prove that if then .

II – Let be the operator defined on which assigns to the function where is the solution of

4. Let . Prove that maps to and that is a convex closed subset of .

5. Prove that for every in we have

6. Prove that if we denote by the solutions of corresponding to we have for every that

7. Prove that is a contraction from to and that has a unique solution.

III Choose bounded in , and denote the solution of which corresponds to .

8. Prove that if , then . (we can prove that ).

9. Prove that and are bounded in .

10. Prove that is compact in for every ball of .

11. Prove that, up to a subsequence, we ca pass to the limit in the sense of distributions in with as .

Uniqueness and Error estimates via Kinetic Entropy Defect Measure

Here are a few thoughts from my preparation for the exam of Kinetic Equations at Universite de Savoie, France. The teachers of the course were Christian Bourdarias and Stephane Gerbi. I had to study an article of Benoit Perthame entitled Uniqueness and Error estimates in First Order Quasilinear Conservation Laws via the Kinetic Entropy Defect Measure.

This was a very nice article to study, since it used many things like distribution theory, measures and regularization. It showed the power of these tools, and motivated me to learn more about them.

As the title of the article says, a relatively new proof of the uniqueness of the solution for a scalar conservation law coupled with some entropy inequalities is given. The only known proof at the time the article was published was due to Kruzkov and was more intricate and difficult to understand than the one provided in the article. The estimates on the entropy defect measure, which will be introduced can yield some error term approximation for approximate equation, which in particular imply unicity at once.

Here are my detailed notes on the article. They are handwritten, but I think they are readable. Perthame-Uniqueness and Error Estimates

Traian Lalescu Student Contest 2011 Problem 4

Let and fixed.

1) Prove that if and only if there exists such that .

2) Prove that then there is a unique function such that .

Weakly harmonic implies harmonic

Suppose is weakly harmonic on an open set , i.e. the relation holds for all . Show that is harmonic in .

PHD Iowa (6202)

See this blog post for a proof.

Divergent integral

Let satisfy on . Show that the integral is convergent if and only if .
PHD 6201

Compacts convergence implies characteristic convergence

Prove that if the sequence of of open sets converges to in the compact way, and if the boundary of is of measure zero, then converges in the way of characteristic functions.

Frontier convergence

Suppose that a sequence of open sets converges in the way of compact sets to an open set . Is it true that we nave ?

Same question for the convergence in the way of characteristic functions and for the Hausdorff convergence.

Open sets convergence

Consider the sequence of open sets of defined by
a) .

b) .

Study the convergence of the sets in the following types of convergence:

• weak convergence of characteristic functions;
• convergence in the topology defined by the Hausdorff distance;
• convergence in the way of compact sets.

(Each of the previous types of convergence will be presented below.)

Harmonic function

Let be semilines on the plane starting from a common point. Prove that if there doesn’t exist any function harmonic on the whole plane that vanishes on the set , then there exists a pair such that there is no function harmonic on the whole plane such that vanishes on .
Miklos Schweitzer 2001