72 research outputs found
A Polygonal Discontinuous Galerkin Method with Minus One Stabilization
We propose an Hybridized Discontinuous Galerkin method on polygonal
tessellations, stabilized by penalizing, locally in each element , a
residual term involving the fluxes, measured in the norm of the dual of
. The scalar product corresponding to such a norm is numerically
realized via the introduction of a (minimal) auxiliary space of VEM type.
Stability and optimal error estimates in the broken norm are proven under
a weak shape regularity assumption allowing the presence of very small edges.
The results of numerical tests confirm the theoretical estimates.Comment: 27 pages, 2 figure
Best approximation results and essential boundary conditions for novel types of weak adversarial network discretizations for PDEs
In this paper, we provide a theoretical analysis of the recently introduced
weakly adversarial networks (WAN) method, used to approximate partial
differential equations in high dimensions. We address the existence and
stability of the solution, as well as approximation bounds. More precisely, we
prove the existence of discrete solutions, intended in a suitable weak sense,
for which we prove a quasi-best approximation estimate similar to Cea's lemma,
a result commonly found in finite element methods. We also propose two new
stabilized WAN-based formulas that avoid the need for direct normalization.
Furthermore, we analyze the method's effectiveness for the Dirichlet boundary
problem that employs the implicit representation of the geometry. The key
requirement for achieving the best approximation outcome is to ensure that the
space for the test network satisfies a specific condition, known as the inf-sup
condition, essentially requiring that the test network set is sufficiently
large when compared to the trial space. The method's accuracy, however, is only
determined by the space of the trial network. We also devise a pseudo-time
XNODE neural network class for static PDE problems, yielding significantly
faster convergence results than the classical DNN network.Comment: 30 pages, 7 figure
FETI-DP for the three-dimensional Virtual Element Method
We deal with the Finite Element Tearing and Interconnecting Dual Primal
(FETI-DP) preconditioner for elliptic problems discretized by the virtual
element method (VEM). We extend the result of [16] to the three dimensional
case. We prove polylogarithmic condition number bounds, independent of the
number of subdomains, the mesh size, and jumps in the diffusion coefficients.
Numerical experiments validate the theoryComment: 28 page
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