Computer Methods in Applied Mechanics and Engineering
Two classes of mixed finite element methods
Computer Methods in Applied Mechanics and Engineering
Least-Squares Finite Element Method for the Stokes Problem with Zero Residual of Mass Conservation
SIAM Journal on Numerical Analysis
Issues Related to Least-Squares Finite Element Methods for the Stokes Equations
SIAM Journal on Scientific Computing
Least-squares spectral elements applied to the Stokes problem
Journal of Computational Physics
Quadrilateral H(div) Finite Elements
SIAM Journal on Numerical Analysis
Stabilization of Low-order Mixed Finite Elements for the Stokes Equations
SIAM Journal on Numerical Analysis
Spectral Discretization of the Vorticity, Velocity, and Pressure Formulation of the Stokes Problem
SIAM Journal on Numerical Analysis
Mass- and Momentum Conservation of the Least-Squares Spectral Element Method for the Stokes Problem
Journal of Scientific Computing
Discrete differential forms for computational modeling
SIGGRAPH '05 ACM SIGGRAPH 2005 Courses
Journal of Computational Physics
Whitney Forms of Higher Degree
SIAM Journal on Numerical Analysis
Isogeometric Discrete Differential Forms in Three Dimensions
SIAM Journal on Numerical Analysis
A locally conservative mimetic least-squares finite element method for the stokes equations
LSSC'09 Proceedings of the 7th international conference on Large-Scale Scientific Computing
Mimetic least-squares spectral/hp finite element method for the poisson equation
LSSC'09 Proceedings of the 7th international conference on Large-Scale Scientific Computing
Journal of Computational Physics
High order geometric methods with exact conservation properties
Journal of Computational Physics
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In this paper we apply the recently developed mimetic discretization method to the mixed formulation of the Stokes problem in terms of vorticity, velocity and pressure. The mimetic discretization presented in this paper and in Kreeft et al. [51] is a higher-order method for curvilinear quadrilaterals and hexahedrals. Fundamental is the underlying structure of oriented geometric objects, the relation between these objects through the boundary operator and how this defines the exterior derivative, representing the grad, curl and div, through the generalized Stokes theorem. The mimetic method presented here uses the language of differential k-forms with k-cochains as their discrete counterpart, and the relations between them in terms of the mimetic operators: reduction, reconstruction and projection. The reconstruction consists of the recently developed mimetic spectral interpolation functions. The most important result of the mimetic framework is the commutation between differentiation at the continuous level with that on the finite dimensional and discrete level. As a result operators like gradient, curl and divergence are discretized exactly. For Stokes flow, this implies a pointwise divergence-free solution. This is confirmed using a set of test cases on both Cartesian and curvilinear meshes. It will be shown that the method converges optimally for all admissible boundary conditions.