progfree++
- 1. UNIVERSITÀ DEGLI STUDI DI ROMA TOR VERGATA Dipartimento di Matematica Laurea in Matematica pura e applicata Laurea in Mathematical Engineering Laboratorio di calcolo Finite Element Method with Freefem++ Filingeri Giuseppe Peluso Fabio Zenelanska Alexandra Prof.ssa Pelosi Francesca Prof. Baldi Paolo 1
- 2. Contents 1 Introduction 3 2 Case study 4 2.1 Poisson problem in (0, 1)2 . . . . . . . . . . . . . . . . . . . . 4 2.2 Poisson problem in the circle . . . . . . . . . . . . . . . . . . . 6 2
- 3. 1 Introduction In mathematics, a partial dierential equation (PDE) is a dierential equation that contains unknown multivariable functions and their partial derivatives. (A special case are ordinary dierential equations (ODEs), which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. In this paper we will consider a time independent PDE given by a 2-dimensional Poisson problem with Dirichlet boundary condition. −∆u = f on Ω u = g on ∂Ω where as usually we denote ∆ the Laplacian operator and f ∈ L2 (Ω). In most general case a PDE closed form solution in not quite easy to obtain. For this reason several discretization method were developed such as Finite Dierence Method (FDM) and Finite Element one (FEM). According to FEM, strong formulation could be redened weakly applying test functions in Sobolev space. Thanks to Divergence Theorem and Green one thus we obtain weak formulation for our problem: Ω u · vdΩ = Ω fvdΩ with test function v choosen in {v : Ω → R|v, ∂v ∂xi ∈ L2 (Ω), v = 0 su ∂Ω}. The above term, seen as bilinear form a(u, v) and a linear function F(v) can be espressed as a(u, v) = F(v) Thus, considering the approximated solution as the right solution for every basis function in the space ϕi, we obtain a(ui, ϕi) = F(ϕi) from which we obtain the linear system Au = f for every v belonging to the space above dened where u is still the solution we are looking for. 3
- 4. 2 Case study Using Freefem++ to approximate the Poisson problem according to Galerkin method with C0 the space of linear stepwise function. In our case, recalling that in −∆u = f on Ω u = g on ∂Ω the equation is not homogeneus, the formulation becomes: Ω u · vdΩ − dΩ ∂u ∂n vdγ = Ω fvdΩ 2.1 Poisson problem in (0, 1)2 let us dene Ω = (0, 1)2 , g(x, y) = y cos(2πx) on ∂Ω as the right solution u(x, y). f(x, y) = 4(π2 )y cos(2πx) as −∆U(x, y) In this case the set Ω is the square (0, 1)2 thus boundaries are splitted in four region. In the Freefem++ code we use triangulated mesh with poly- nomials P1. Then, setting laplace in the source code the program solves weak formulation problem generating u function isovalued plot and its trian- gulated domain Th. Finally, we obtain the discretized surface for dierent number of points discretization. The error behaviour in L2 norm and the order of convergence are the following: 4
- 5. (a) n = 4 (b) n = 8. (c) n = 16. (d) n = 32. Figure 1: Poisson discretization for dierent number of mesh points 5
- 6. 2.2 Poisson problem in the circle let us dene Ω = (x, y) ∈ R2 : x2 + y2 9, g(x, y) = x2 + y2 − 9 on ∂Ω as the right solution u(x, y), f(x, y) = −4 as −∆U(x, y) In this case the set Ω is the circle of radius 3 thus boundaries are equal. In the Freefem++ code we use triangulated mesh with polynomials P1. Then, setting laplace in the source code the program solves weak formulation problem generating u function isovalued plot and its triangulated domain Th. Finally, we obtain the discretized surface for dierent number of points discretization. Per il problema i) b otteniamo i risultati seguenti. In questo caso, essendo il bordo un cerchio, viene parametrizzato. The error behaviour in L2 norm and the order of convergence are the fol- lowing: 6
- 7. (a) n = 4 (b) n = 8. (c) n = 16. (d) n = 32. Figure 2: Poisson discretization for dierent number of mesh points 7
- 8. Considering another polynomial space for basis function with another one of higher order we expect a higher order of accuracy and a lower error in L2 norm for xed number of discretization points. Using polynomial belong- ing to P2, thus the error decrease for xed mesh points. Note that in the second case the error does not decrease since we are near to the maximum representation error in C++. 8