30 research outputs found
Primal dual mixed finite element methods for indefinite advection--diffusion equations
We consider primal-dual mixed finite element methods for the
advection--diffusion equation. For the primal variable we use standard
continuous finite element space and for the flux we use the Raviart-Thomas
space. We prove optimal a priori error estimates in the energy- and the
-norms for the primal variable in the low Peclet regime. In the high
Peclet regime we also prove optimal error estimates for the primal variable in
the norm for smooth solutions. Numerically we observe that the method
eliminates the spurious oscillations close to interior layers that pollute the
solution of the standard Galerkin method when the local Peclet number is high.
This method, however, does produce spurious solutions when outflow boundary
layer presents. In the last section we propose two simple strategies to remove
such numerical artefacts caused by the outflow boundary layer and validate them
numerically.Comment: 25 pages, 6 figures, 5 table
Improved ZZ A Posteriori Error Estimators for Diffusion Problems: Conforming Linear Elements
In \cite{CaZh:09}, we introduced and analyzed an improved Zienkiewicz-Zhu
(ZZ) estimator for the conforming linear finite element approximation to
elliptic interface problems. The estimator is based on the piecewise "constant"
flux recovery in the conforming finite element space. This
paper extends the results of \cite{CaZh:09} to diffusion problems with full
diffusion tensor and to the flux recovery both in piecewise constant and
piecewise linear space.Comment: arXiv admin note: substantial text overlap with arXiv:1407.437
Residual-based a posteriori error estimation for immersed finite element methods
In this paper we introduce and analyze the residual-based a posteriori error
estimation of the partially penalized immersed finite element method for
solving elliptic interface problems. The immersed finite element method can be
naturally utilized on interface-unfitted meshes. Our a posteriori error
estimate is proved to be both reliable and efficient with reliability constant
independent of the location of the interface. Numerical results indicate that
the efficiency constant is independent of the interface location and that the
error estimation is robust with respect to the coefficient contrast
An a posteriori error estimate of the outer normal derivative using dual weights
We derive a residual based a-posteriori error estimate for the outer normal
flux of approximations to {the diffusion problem with variable coefficient}. By
analyzing the solution of the adjoint problem, we show that error indicators in
the bulk may be defined to be of higher order than those close to the boundary,
which lead to more economic meshes. The theory is illustrated with some
numerical examples.Comment: 27 pages, 13 figures, 3 table
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
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