2 research outputs found
Asymptotic preserving scheme for anisotropic elliptic equations with deep neural network
In this paper, a new asymptotic preserving (AP) scheme is proposed for the
anisotropic elliptic equations. Different from previous AP schemes, the actual
one is based on first-order system least-squares for second-order partial
differential equations, and it is uniformly well-posed with respect to
anisotropic strength. The numerical computation is realized by a deep neural
network (DNN), where least-squares functionals are employed as loss functions
to determine parameters of DNN. Numerical results show that the current AP
scheme is easy for implementation and is robust to approximate solutions or to
identify anisotropic strength in various 2D and 3D tests
Preserving the accuracy of numerical methods discretizing anisotropic elliptic problems
In this paper we study the loss of precision of numerical methods
discretizing anisotropic problems and propose alternative approaches free from
this drawback. The deterioration of the accuracy is observed when the
coordinates and the mesh are unrelated to the anisotropy direction. While this
issue is commonly addressed by increasing the scheme approximation order, we
demonstrate that, though the gains are evident, the precision of these
numerical methods remain far from optimal and limited to moderate anisotropy
strengths. This is analysed and explained by an amplification of the
approximation error related to the anisotropy strength. We propose an approach
consisting in the introduction of an auxiliary variable aimed at removing the
amplification of the discretization error. By this means the precision of the
numerical approximation is demonstrated to be independent of the anisotropy
strength