2.1.1 Domain discretization

The most known technique based on a domain discretization is the FDM where the unknown function is sampled in a finite number of points p1, p2, …, pN ∈ Ω and the derivatives are approximated with finite differences. By writing the differential equation for every p, with the derivative approximated as finite differences, one obtains a system of N equations in N unknowns that can be solved with known techniques. If the original PDE was linear, the discretized system will be linear too.

FDM is maybe the simplest approach and the most intuitive, and it can work quite well for simple problems and geometries. Moreover, in the linear case, since any approximation of a derivative in p will consider only few points around p, the matrix of the discretized linear system will be very sparse, allowing for a reduction in the computational effort. The application of FDM techniques becomes difficult, albeit possible, in the case of complex problems.

Intuitively, better approximations of the solution require finer discretization (the exact meaning of finer depends on the specific approach), especially if the solution has some regions of large variability. Since finer discretization implies larger problems (and, therefore, higher computational efforts), it is of interest to be able to change locally the discretization resolution to the solution variability, possi-

This need for different resolutions in different regions is the idea that links PDE with multiresolution analysis. The birth of multiresolution analysis goes back to 1990 with the works of Mallat [1] and Meyer [2]. Since then there have been a large number of papers ranging from very theoretical ones to application [3]. In a

represented as a nesting of spaces with different levels of "resolution." This allows to write a signal as the sum of a "low-resolution" version plus some higher-

Because of the ability of changing the resolution used to observe the signal (by adding or removing details), the multiresolution analysis is sometimes described as a mathematical zoom. This fact inspired many applications, including numerical solution of PDEs where they sound promising, especially for those problems that contains localized phenomena (e.g., shockwaves) or intra-scale interaction (e.g., turbulence). The objective of this chapter is to introduce the reader to the application of wavelets to PDE solutions. This chapter can be ideally divided in three parts: in the first part, we recall briefly the main concepts about PDE and the main algorithms for solving PDE; successively we do a brief recall of multiresolution analysis and wavelets including also multiwavelets and second-generation wavelets that find often application in PDE solutions; and finally, we will illustrate few techniques that

Ω ⊆ IR<sup>d</sup> is the domain where the functions of interest are defined. The boundary of <sup>Ω</sup> is partitioned as follows: <sup>∂</sup><sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup><sup>D</sup> <sup>∪</sup> <sup>Ω</sup>N, <sup>Ω</sup><sup>D</sup> <sup>∩</sup> <sup>Ω</sup><sup>N</sup> <sup>¼</sup> <sup>∅</sup>, where <sup>Ω</sup><sup>D</sup> or <sup>Ω</sup><sup>N</sup> can be

<sup>H</sup>ð Þ <sup>Ω</sup> will denote a space of functions <sup>u</sup> : <sup>Ω</sup> ! IR defined on <sup>Ω</sup> <sup>⊆</sup>IR<sup>d</sup>.

three large classes: equilibrium, propagation, or eigenvalue problems.

∂u

Most of the physical problems modeled by PDEs fall in one of the following

• In an equilibrium problem, we are interested in finding a function u∈ Hð Þ Ω

In (1)–(3) D is an operator that includes derivatives and f ∈ Hð Þ Ω is known. Boundary conditions are typically given as constraint about u or its derivatives

Du ¼ f in Ω (1) u ¼ uD on Ω<sup>D</sup> (2)

<sup>∂</sup><sup>n</sup> <sup>¼</sup> uN on <sup>Ω</sup><sup>N</sup> (3)

ð Þ IR , but not only) is

multiresolution analysis, a space of signals (most commonly L<sup>2</sup>

bly in an adaptive way.

Wavelet Transform and Complexity

resolution "details."

can be found in the literature.

2. Generalities on PDE

such that

36

IR≥<sup>0</sup> is the set of nonnegative reals.

1.1 Notation

empty.

### 2.1.2 Function space discretization

Another class of techniques discretizes the function space Hð Þ Ω by approximating it with an n-dimensional space Hn, that is, unknown function u is approximated as

$$u \approx \sum\_{i=1}^{n} a\_i b\_i \qquad \quad a\_i \in \mathbb{R} \tag{6}$$

developed [4]. A typical meshless approach is to approximate u with a discrete

I

where XI are a set of points of Ω and ρ is a scale factor that allows to change the "resolution" of kernel function ϕ. Coefficient C<sup>ρ</sup> can be used to keep the energy

After expressing u as linear combination of bi, we are left with the problem of determining the coefficients of the linear combination. Several approaches are possible; the easiest way to briefly present them is by rewriting the differential

If we restrict u to be a linear combination of bi, most probably we will not be able to make residual (10) exactly zero; therefore, we will aim to make it as small as possible. Since the result of the residual operator is a function, there are many

With the collocation approach, we choose a number of points of the domain p1, p2, …, pn ∈ Ω and ask that the residual is zero on the chosen points, that is,

� � � f pj

� � <sup>¼</sup> <sup>∑</sup> n i¼1

Eq. (11) represents a system of n equations having as unknown the coefficients

α<sup>i</sup> Dbi ½ � pj

With reference to Remark 2.1, one can generalize the collocation method by using a set of linear functionals σ<sup>j</sup> : Hð Þ! Ω IR. In this case one can obtain a gener-

Another approach is to solve Ru ¼ 0 in a least square sense, that is, to search

Standard algebra allows to show that (14) is minimized when Ru is orthogonal

αiσjð Þ Dbi |fflfflffl{zfflfflffl} Aj,i

k k Ru

<sup>α</sup>I<sup>ϕ</sup> <sup>x</sup> � xI ρ

� � (9)

Ru ≔ Du � f ¼ 0 (10)

� �. Note that (12) is a linear system in unknowns <sup>α</sup>i.

� � <sup>j</sup> <sup>¼</sup> <sup>1</sup>, …, n (11)

αiAj,i j ¼ 1, …, n (12)

j ¼ 1, …, n (13)

<sup>2</sup> <sup>¼</sup> h i <sup>R</sup>u; <sup>R</sup><sup>u</sup> (14)

u xð Þ¼ C<sup>ρ</sup> ∑

where operator R : Hð Þ! Ω Hð Þ Ω is called the residual.

� � <sup>¼</sup> ½ � <sup>D</sup><sup>u</sup> pj

convolution with kernel as a chosen function φ, that is,

DOI: http://dx.doi.org/10.5772/intechopen.82820

Wavelets for Differential Equations and Numerical Operator Calculus

constant as ρ is changed.

equation as

f pj

Remark 2.2.

� � <sup>¼</sup> <sup>D</sup> <sup>∑</sup>

2.1.3 Exploiting the discretization

possible approaches in minimizing it.

n i¼1 αibi � � pj

where, clearly, Aj,i ¼ Dbi ½ � pj

alized version of (12), namely,

for coefficients α<sup>i</sup> that minimize

to ∂Ru=∂α<sup>i</sup> for every i, that is,

39

0 ¼ ½ � Ru pj

αi, i ¼ 1, …, n. For example, if D is linear, (11) becomes

� � <sup>¼</sup> <sup>∑</sup> n i¼1

σjf ¼ ∑ n i¼1

where f g bi <sup>n</sup> <sup>i</sup>¼<sup>1</sup> is a basis of Hn.

By exploiting approximation (6), one can transform PDE (1)–(3) into a finitedimensional problem. The different solution techniques differ in how (6) is used and in the way of choosing space Hn and its basis f g bi <sup>n</sup> i¼1.

One possibility is to choose functions bi that are infinitely differentiable and nonvanishing on the whole Ω. This gives rise to so-called spectral methods. Typical choices for basis functions can be complex exponential/sinusoidal functions (if the solution is expected to be periodic), Chebyshev polynomials (for separable domains, e.g., d-dimensional cubes), and spherical harmonics (for systems with spherical symmetry). Spectral methods can work very well if the solution is expected to be smooth; they can even converge exponentially fast. However, their spatial localization is not good, and if the functions involved are not smooth (e.g., they are discontinuous), they lose most of their interest.

Another approach, very popular, is FEM that chooses functions bi by first partitioning the domain Ω into a set of elements (triangles and their multidimensional counterpart are a popular choice) and assigning to every element a suitable finite-dimensional vector space. The final approximation of u is constructed in a piecewise fashion by gluing, so to say, the approximations of u over every single element.

In a typical implementation of FEM, all the elements are affine images of a single reference element. This simplifies the implementation since it suffices to choose only the vector space of the reference element T0. Another popular choice is to choose the space associated to the elements as spaces of polynomials. The basis is selected by choosing a set of control points in q1, q2, … ∈ T<sup>0</sup> and choosing as basis vectors bi the polynomials that satisfy the interpolation property

$$b\_i(q\_j) = \delta\_{i,j} = \begin{pmatrix} \mathbf{1} & \text{if } i = j \\ \mathbf{0} & \text{if } i \neq j \end{pmatrix} \tag{7}$$

Remark 2.1 (generalized collocation method).

A generalization of this idea is to choose a set of functionals σ<sup>j</sup> mapping functions defined over T<sup>0</sup> to IR and requiring

$$
\sigma\_j(b\_i) = \delta\_{i,j} \tag{8}
$$

Eq. (8) gives back (7) if σ<sup>j</sup> is defined as the functional that corresponds to evaluating the argument of the functional in qj . Eq. (8) is, however, more general than (7) since it can be used, for example, to control the flow through a face of the element.

An issue with FEM is that creating the grid of elements can be expensive. This is especially true in those problems where the geometry is not fixed but needs to be updated. An example of this type of system is free-surface fluid flows, where the interface between air and fluid changes with time, requiring a continuous update of the mesh. In order to solve this problem, meshless methods have been

developed [4]. A typical meshless approach is to approximate u with a discrete convolution with kernel as a chosen function φ, that is,

$$u(\mathbf{x}) = \mathbf{C}\_{\rho} \sum\_{I} a\_{I} \phi \left( \frac{\mathbf{x} - \mathbf{x}\_{I}}{\rho} \right) \tag{9}$$

where XI are a set of points of Ω and ρ is a scale factor that allows to change the "resolution" of kernel function ϕ. Coefficient C<sup>ρ</sup> can be used to keep the energy constant as ρ is changed.
