We present novel enhanced finite element methods for the Darcy problem starting from the non stable continuous P1/P0 finite element space enriched with multiscale functions. The methods are derived in a Petrov-Galerkin framework where both velocity and pressure trial spaces are enriched with functions based on residuals of the strong equations in each element and edge partition. The strategy le...

We propose a nonconforming finite element method for isentropic viscous gas flow in situations where convective effects may be neglected. We approximate the continuity equation by a piecewise constant discontinuous Galerkin method. The velocity (momentum) equation is approximated by a finite element method on div–curl form using the nonconforming Crouzeix– Raviart space. Our main result is that...

Stability analyses and error estimates are carried out for a number of commonly used numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown that all the schemes we considered are either unconditionally energy stable, or conditionally energy stable with reasonable stability conditions in the semi-discretized versions. Error estimates for selected schemes with a spectral-Ga...

A Discontinuous Galerkin (DG) method with solenoidal approximation for the simulation of incompressible flow is proposed. It is applied to the solution of the Stokes equations. The Interior Penalty Method is employed to construct the DG weak form. For every element, the approximation space for the velocity field is decomposed as direct sum of a solenoidal space and an irrotational space. This a...

In this paper, our purpose is to study the classical continuous optimal control (CCOC) for quaternary nonlinear parabolic boundary value problems (QNLPBVPs). The existence and uniqueness theorem (EUTh) state vector solution (QSVS) of weak form (WF) QNLPBVPs with a given (QCCCV) stated proved via Galerkin Method (GM) first compactness under suitable assumptions(ASSUMS). Furthermore, continuity o...

A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for b...

Abstract. A C0-weak Galerkin (WG) method is introduced and analyzed for solving the biharmonic equation in 2D and 3D. A weak Laplacian is defined for C0 functions in the new weak formulation. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established in both a discrete H2 norm and the L2 norm, for the weak Galerkin finite...

Standard weak solutions to the Poisson problem on a bounded domain have squareintegrable derivatives, which limits the admissible regularity of inhomogeneous data. The concept of solution may be further weakened in order to define solutions when data is rough, such as for inhomogeneous Dirichlet data that is only square-integrable over the boundary. Such very weak solutions satisfy a nonstandar...