1,724 research outputs found
A simple multigrid scheme for solving the Poisson equation with arbitrary domain boundaries
We present a new multigrid scheme for solving the Poisson equation with
Dirichlet boundary conditions on a Cartesian grid with irregular domain
boundaries. This scheme was developed in the context of the Adaptive Mesh
Refinement (AMR) schemes based on a graded-octree data structure. The Poisson
equation is solved on a level-by-level basis, using a "one-way interface"
scheme in which boundary conditions are interpolated from the previous coarser
level solution. Such a scheme is particularly well suited for self-gravitating
astrophysical flows requiring an adaptive time stepping strategy. By
constructing a multigrid hierarchy covering the active cells of each AMR level,
we have designed a memory-efficient algorithm that can benefit fully from the
multigrid acceleration. We present a simple method for capturing the boundary
conditions across the multigrid hierarchy, based on a second-order accurate
reconstruction of the boundaries of the multigrid levels. In case of very
complex boundaries, small scale features become smaller than the discretization
cell size of coarse multigrid levels and convergence problems arise. We propose
a simple solution to address these issues. Using our scheme, the convergence
rate usually depends on the grid size for complex grids, but good linear
convergence is maintained. The proposed method was successfully implemented on
distributed memory architectures in the RAMSES code, for which we present and
discuss convergence and accuracy properties as well as timing performances.Comment: 33 pages, 15 figures, accepted for publication in Journal of
Computational Physic
A Comparison Study of Two Methods for Elliptic Boundary Value Problems
In this paper, we perform a comparison study of two methods (the embedded
boundary method and several versions of the mixed finite element method) to
solve an elliptic boundary value problem
Solving elliptic problems with discontinuities on irregular domains – the Voronoi Interface Method.
We introduce a simple method, dubbed the Voronoi Interface Method, to solve Elliptic problems with discontinuities across the interface of irregular domains. This method produces a linear system that is symmetric positive definite with only its right-hand-side affected by the jump conditions. The solution and the solution's gradients are second-order accurate and first-order accurate, respectively, in the L∞L∞ norm, even in the case of large ratios in the diffusion coefficient. This approach is also applicable to arbitrary meshes. Additional degrees of freedom are placed close to the interface and a Voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach. Both the locations of the additional degrees of freedom and their Voronoi discretizations are straightforward in two and three spatial dimensions
A cusp-capturing PINN for elliptic interface problems
In this paper, we propose a cusp-capturing physics-informed neural network
(PINN) to solve discontinuous-coefficient elliptic interface problems whose
solution is continuous but has discontinuous first derivatives on the
interface. To find such a solution using neural network representation, we
introduce a cusp-enforced level set function as an additional feature input to
the network to retain the inherent solution properties; that is, capturing the
solution cusps (where the derivatives are discontinuous) sharply. In addition,
the proposed neural network has the advantage of being mesh-free, so it can
easily handle problems in irregular domains. We train the network using the
physics-informed framework in which the loss function comprises the residual of
the differential equation together with certain interface and boundary
conditions. We conduct a series of numerical experiments to demonstrate the
effectiveness of the cusp-capturing technique and the accuracy of the present
network model. Numerical results show that even using a one-hidden-layer
(shallow) network with a moderate number of neurons and sufficient training
data points, the present network model can achieve prediction accuracy
comparable with traditional methods. Besides, if the solution is discontinuous
across the interface, we can simply incorporate an additional supervised
learning task for solution jump approximation into the present network without
much difficulty
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