FEM and BEM simulations with the Gypsilab framework

Abstract

Gypsilab is a Matlab framework which aims at simplifying the development of numerical methods that apply to the resolution of problems in multiphysics, in particular, those involving FEM or BEM simulations. The peculiarities of the framework, with a focus on its ease of use, are shown together with the methodology that have been followed for its development. Example codes that are short though representative enough are given both for FEM and BEM applications. A performance comparison with FreeFem++ is provided, and a particular emphasis is made on problems in acoustics and electromagnetics solved using the BEM and for which compressed H-matrices are used.

New preconditioners for the Laplace and Helmholtz integral equations on open curves: analytical framework and numerical results

“…All the simulations are run on a personal laptop with an eight-core intel i7 processor, a clock rate of 2.8GHz, and 16GB of RAM. The method is implemented in the language Matlab R2018a, and uses the environment Gypsilab 1 developed by Matthieu Aussal and François Alouges [3]. The full code of our method is freely available online 2 .…”

“…We verify those rates numerically. For the Dirichlet problem, we solve two test cases S 0,ω α 1 = u 1 and S 0,ω α 2 = u 2 having the explicit solutions α 1 (x) = ω(x) and α 2 = ω(x) 3 , for adequately chosen right hand sides (rhs) u 1 and u 2 . One can check that α 1 ∈ T s for s < 3 2 and α 1 / ∈ T 3/2 , while α 2 ∈ T 2 .…”

Data completion method for the Helmholtz equation via surface potentials for partial Cauchy data

“…One can see that if (f, g) is known, u can be computed in the entire domain Ω using (2). Applying trace formulas for layer potentials to (2), one can derive a relation between (f 1 , g 1 ), which are the known data, and the unknown data (f 2 , g 2 ).…”

“…Therefore, it may be interesting to use the reconstructed trace on ∂Ω (for which we observed a much higher accuracy) to solve a Dirichlet problem for the Helmholtz equation inside Ω then compute the normal derivative from the computed solution. These two steps can be done using integral representation formulas (2). Taking the normal trace in (2) we obtain…”

“…In the example we assume that the domain Ω is a ball of radius 1 and the data is available on Γ 2 = {(x, y, z) ∈ ∂Ω; x > −0.5}. Whence the Cauchy data is reconstructed on Γ 1 leading to a knowledge of u δ Γ and ∂ ν u δ Γ , we evaluate the field and the normal derivative on the inner surface using the integral representation (2). We observe that the accuracy obtained on the inner surface is very good as shown in Table 6.…”