We consider the numerical solution of a steady-state diffusion problem where the diffusion coefficient is the exponent of a random field. The standard stochastic Galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it. We consider a reformulated version of this problem as a stochastic convection-diffusion problem with...

We introduce the concept of data-driven finite element methods. These are finite-element discretizations partial differential equations (PDEs) that resolve quantities interest with striking accuracy, regardless underlying mesh size. The methods obtained within a machine-learning framework during which parameters defining method tuned against available training data. In particular, we use stable...

Abstract: In this paper, by using the Galerkin method and the generalized Brouwer’s theorem, some problems of the higher eigenvalues are studied for a class of singular quasiliner elliptic equations in the weighted Sobolev spaces. The existence of weak solutions is obtained for this problem.

We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to the Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smoo...

We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth b...

The Cahn-Hilliard equation is discretized by a Galerkin finite element method based on continuous piecewise linear functions in space and discontinuous piecewise constant functions in time. A posteriori error estimates are proved by using the methodology of dual weighted residuals.

We consider the Galerkin nite element method for partial diiferential equations in two dimensions, where the nite-dimensional space used consists of piecewise (isoparametric) polynomials enriched with bubble functions. Writing L for the diierential operator, we show that for elliptic convection-diiusion problems, the component of the bubble enrichment that stabilizes the method is equivalent to...

A local a priori and a posteriori analysis is developed for the Galerkin method with discontinuous finite elements for solving stationary diffusion problems. The main results are an optimal-order estimate for the point-wise error and a corresponding a posteriori error bound. The proofs are based on weighted -norm error estimates for discrete Green functions as already known for the ‘continuous’...