14 research outputs found
Convergence Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods
In this article, we prove convergence of the weakly penalized adaptive
discontinuous Galerkin methods. Unlike other works, we derive the contraction
property for various discontinuous Galerkin methods only assuming the
stabilizing parameters are large enough to stabilize the method. A central idea
in the analysis is to construct an auxiliary solution from the discontinuous
Galerkin solution by a simple post processing. Based on the auxiliary solution,
we define the adaptive algorithm which guides to the convergence of adaptive
discontinuous Galerkin methods
Convergence of an adaptive mixed finite element method for general second order linear elliptic problems
The convergence of an adaptive mixed finite element method for general second
order linear elliptic problems defined on simply connected bounded polygonal
domains is analyzed in this paper. The main difficulties in the analysis are
posed by the non-symmetric and indefinite form of the problem along with the
lack of the orthogonality property in mixed finite element methods. The
important tools in the analysis are a posteriori error estimators,
quasi-orthogonality property and quasi-discrete reliability established using
representation formula for the lowest-order Raviart-Thomas solution in terms of
the Crouzeix-Raviart solution of the problem. An adaptive marking in each step
for the local refinement is based on the edge residual and volume residual
terms of the a posteriori estimator. Numerical experiments confirm the
theoretical analysis.Comment: 24 pages, 8 figure
Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem
In this paper, we analyze the convergence and optimality of a standard
adaptive nonconforming linear element method for the Stokes problem. After
establishing a special quasi--orthogonality property for both the velocity and
the pressure in this saddle point problem, we introduce a new prolongation
operator to carry through the discrete reliability analysis for the error
estimator. We then use a specially defined interpolation operator to prove
that, up to oscillation, the error can be bounded by the approximation error
within a properly defined nonlinear approximate class. Finally, by introducing
a new parameter-dependent error estimator, we prove the convergence and
optimality estimates
Recurrent Neural Networks as Optimal Mesh Refinement Strategies
We show that an optimal finite element mesh refinement algorithm for a
prototypical elliptic PDE can be learned by a recurrent neural network with a
fixed number of trainable parameters independent of the desired accuracy and
the input size, i.e., number of elements of the mesh. Moreover, for a general
class of PDEs with solutions which are well-approximated by deep neural
networks, we show that an optimal mesh refinement strategy can be learned by
recurrent neural networks. This includes problems for which no optimal adaptive
strategy is known yet