In the same space, the coordinate surface r = 1 in spherical coordinates is the surface of a unit sphere, which is curved. A Cartesian coordinate surface in this space is a coordinate plane for example z = 0 defines the x- y plane. Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space ( R 3) are cylindrical and spherical coordinates. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. Curvilinear (top), affine (right), and Cartesian (left) coordinates in two-dimensional space Zlámal, Curved elements in the finite element method. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations, Math. Verfürth, Finite element approximation of incompressible Navier-Stokes equations with slip boundary condition, Numer. thesis, Massachusetts Institute of Technology, 1973. Scott, Finite-Element Techniques for Curved Boundaries, Ph.D. Comput., Oxford University Press, New York, 2003. Monk, Finite Element Methods for Maxwell's Equations, Numer. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comput. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Taylor, Extension and representation of divergence-free vector fields on bounded domains, Math. Sharma, Divergence-conforming discontinuous Galerkin methods and $C^0$ interior penalty methods, SIAM J. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev. Neilan, Inf-sup stable finite elements on barycentric refinements producing divergence-free approximations in arbitrary dimensions, SIAM J. Neilan, Exact smooth piecewise polynomial sequences on Alfeld splits, Math. Urquiza, Penalty: Finite element approximation of Stokes equations with slip boundary conditions, Numer. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
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DIVERGENCE IN CURVED SPACE FREE
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