Numerical solution to partial differential equations using fractals

The goal of this work is to construct and use smooth fractal functions to find a numerical solution of elliptic partial differential equations in one dimension. We focus on univariate R−valued fractal functions defined on interval [0, 1] which are C0 smooth. These fractal functions provide more flexibility for an approximation and interpolation. They are used as shape functions in the concept of hierarchical basis in the finite element method. Based on a continuous RB operator defined on a space of bounded continuous functions, a fractal bubble is initially constructed on the interval [0, 1]. Further, linear nodal basis functions defined on the interval [0, 1] are expressed as fixed points of an iterated function system. Consequently, they are established as C0−smooth fractal functions. Thus, a hierarchical fractal basis is constructed on the interval [0, 1], consisting of three shape functions: two linear nodal basis functions and a hierarchical element namely, the fractal bubble.

The nodal basis functions ensure the continuity at connecting nodes while the fractal bubble yields additional degree of freedom without increasing the numbers of nodes in a mesh. Subsequently, a finite element mesh is constructed by discretizing the interval [0, 1] in 2n elements. The hierarchical fractal basis on a finite element mesh is then obtained by translating and dilating the three hierarchical basis functions. This basis defined on the finite element mesh spans the space of fractal functions and is used as a finite element space. This space has a direct sum hierarchical decomposition into a space of piecewise linear nodal basis functions and a space of piecewise fractal functions which are zero at the nodes. Subsequently, to set up element matrices in a finite element model, the fractal functions are integrated explicitly using the self-similarity property. However, Trapezoidal rule is used for numerical integration of fractal function where the explicit integration of 1 fractal functions is unattainable.

As an illustration, a numerical solution to the Poisson equation with homogeneous Dirichlet’s boundary conditions is found. A numerical error is computed in L1− norm, L2− norm. A bound of an interpolation error in L2 norm and H − 1 seminorm is established theoretically. A python code is used for generating the fractal bubble from an iterated function system and for computational purpose to find an approximate solution to Poisson equation in one dimension.