Conservation of mass and momentum of the least-squares spectral collocation scheme for the Stokes problem

A locally conservative, discontinuous least‐squares finite element method for the Stokes equations

“…(12). Assume that the exact solution of the VVP Stokes system (8) is such that u ∈ H r +1 ( ), ∈ H r ( ) and p ∈ H r ( ). There exists a constant C>0 such that…”

“…There are several first-order formulations of the Stokes equations [1, Section 7.1]. The most widely used is the VVP first-order system ∇ × +∇ p = f on −∇ ×u = 0 on ∇ ·u = 0 on (8) which is derived from (5) by using the vorticity = ∇ ×u as a new dependent variable and applying the identity ∇ ×∇ ×u = − u+∇(∇ ·u) to rewrite the momentum equation in terms of the vorticity. The VVP system is augmented with the velocity boundary condition (6) and the zero mean constraint (7).…”

“…Moreover, a method can be constructed that combines conventional LSFEM with a correction term computed by the so-called L L * [10] least-squares approach [7]. Other remedies include alternative formulations of the governing equations obtained by adding new variables and corresponding boundary conditions [5,6], high order [11] or spectral [8,9] elements, or weighting the continuity equation more strongly [4]. Compatible finite element spaces are another approach to improved mass conservation.…”

“…However, this changed the least-squares principle to a saddle-point problem, thereby negating one of the key advantages of LSFEMs over other methods. Subsequently, mass conservation in least-squares methods for the Stokes and the Navier-Stokes equations has been studied extensively in the literature [2][3][4][5][6][7][8][9].…”

A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation

“…A significant advantage of the spectral methods over the local methods is the high accuracy of spectral techniques [12][13][14][15][16][17]. Spectral collocation method was proposed for approximating nonlinear differential equations [18][19][20], integral equations [21,22], integro-differential equations [23,24], fractional orders differential equations [25], function approximation and variational problems [26].…”