1,209 research outputs found
Some Error Analysis on Virtual Element Methods
Some error analysis on virtual element methods including inverse
inequalities, norm equivalence, and interpolation error estimates are presented
for polygonal meshes which admits a virtual quasi-uniform triangulation
A mesh-free method using piecewise deep neural network for elliptic interface problems
In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the interface, we employ different neural networks for each sub-domain. By reformulating the interface problem as a least-squares problem, we discretize the objective function using mean squared error via sampling and solve the proposed deep least-squares method by standard training algorithms such as stochastic gradient descent. The discretized objective function utilizes only the point-wise information on the sampling points and thus no underlying mesh is required. Doing this circumvents the challenging meshing procedure as well as the numerical integration on the complex interfaces. To improve the computational efficiency for more challenging problems, we further design an adaptive sampling strategy based on the residual of the least-squares function and propose an adaptive algorithm. Finally, we present several numerical experiments in both 2D and 3D to show the flexibility, effectiveness, and accuracy of the proposed deep least-square method for solving interface problems
An Arbitrarily High Order Unfitted Finite Element Method for Elliptic Interface Problems with Automatic Mesh Generation
We consider the reliable implementation of high-order unfitted finite element
methods on Cartesian meshes with hanging nodes for elliptic interface problems.
We construct a reliable algorithm to merge small interface elements with their
surrounding elements to automatically generate the finite element mesh whose
elements are large with respect to both domains. We propose new basis functions
for the interface elements to control the growth of the condition number of the
stiffness matrix in terms of the finite element approximation order, the number
of elements of the mesh, and the interface deviation which quantifies the mesh
resolution of the geometry of the interface. Numerical examples are presented
to illustrate the competitive performance of the method.Comment: 34 pages, 20 figure
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