The penalty–projection method for the solution of Navier–Stokes equations may be viewed as a projection scheme where an augmentation term is added in the first stage, namely the solution of the momentum balance equation, to constrain the divergence of the predicted velocity field. After a presentation of the scheme in the time semi-discrete formulation, then in fully discrete form for a finite ...

The model for pricing of American option gives rise to a parabolic variational inequality. We first use penalty function approach to reformulate it as an equality problem. Since the problem is defined on an unbounded domain, we truncate it to a bounded domain and discuss error due to truncation and penalization. Finite element method is then applied to the penalized problem on the truncated dom...

A continuous interior penalty hp-finite element method that penalizes the jump of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for first-order and advection-dominated transport operators. The analysis relies on three technical results that are of independent interest: an hp-inverse trace inequality, a local discontinuous to continuous hp-interpolation...

We consider a finite element method with symmetric stabilisation for the discretisation of transient convection--diffusion equation. For time-discretisation we either second order backwards differentiation formula or Crank-Nicolson method. Both convection term and associated are treated explicitly using an extrapolated approximate solution. prove stability $\tau^2 + h^{p+{\frac12}}$ error estim...

This paper presents a comparative study on the newly introduced weak Galerkin finite element methods (WGFEMs) with the widely accepted discontinuous Galerkin finite element methods (DGFEMs) and the classical mixed finite element methods (MFEMs) for solving second-order elliptic boundary value problems. We examine the differences, similarities, and connection among these methods in scheme formul...

In this work we present a theoretical analysis for a residual-type error estimator for locally conservative mixed methods. This estimator was first introduced by Braess and Verfürth for the Raviart–Thomas mixed finite element method working in mesh-dependent norms. We improve and extend their results to cover any locally conservative mixed method under minimal assumptions, in particular, avoidi...

Several stabilized finite element methods for the Stokes eigenvalue problem based on the lowest equal-order finite element pair are numerically investigated. They are penalty, regular, multiscale enrichment, and local Gauss integration method. Comparisons between them are carried out, which show that the local Gauss integration method has good stability, efficiency, and accuracy properties, and...