Augmented high order finite volume element method for elliptic PDEs in non-smooth domains: Convergence study

Augmented high order finite volume element method for elliptic PDEs in non-smooth domains: Convergence study

1. Background The Finite Volume ElementMethod (FVEM) is a numerical method for approximating the solution of a partial differential equation (PDE) in a trial function space spanned by piecewise polynomial basis functions, similar to the Finite Element Method (FEM). The coefficients of the linear combination of the basis functions are obtained by imposing the PDE through integrations over contro...

Error Estimates for a Finite Volume Element Method for Elliptic PDEs in Nonconvex Polygonal Domains

We consider standard finite volume piecewise linear approximations for second order elliptic boundary value problems on a nonconvex polygonal domain. Based on sharp shift estimates, we derive error estimations in H1-, L2and L∞-norms, taking into consideration the regularity of the data. Numerical experiments and counterexamples illustrate the theoretical results.

Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs

We prove convergence of adaptive finite element methods (AFEM) for general (nonsymmetric) second order linear elliptic PDE, thereby extending the result of Morin et al [6, 7]. The proof relies on quasi-orthogonality, which accounts for the bilinear form not being a scalar product, together with novel error and oscillation reduction estimates, which now do not decouple. We show that AFEM is a co...

High-Order Multiscale Finite Element Method for Elliptic Problems

In this paper, a new high-order multiscale finite element method is developed for elliptic problems with highly oscillating coefficients. The method is inspired by the multiscale finite element method developed in [3], but a more explicit multiscale finite element space is constructed. The approximation space is nonconforming when oversampling technique is used. We use a PetrovGalerkin formulat...

The Finite Element Method for 2D elliptic PDEs