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By MacKenzie, D.N.

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Then ϕ is a homeomorphism of Ω onto Ω. If we have additional conditions on the boundary ∂Ω of the domain Ω we can avoid the condition that the Jacobian is positive at least at one point of the boundary. 4 Suppose that ϕ maps the boundary ∂Ω of the domain Ω homeomorphically onto itself, boundary ∂Ω is a n−1-dimensional manifold of class C1 and that det ϕ (x) > 0 (x ∈ Ω), Then ϕ is a homeomorphism of Ω onto Ω. Consider mappings from Ω0 onto Ω. 5 Suppose that ϕ maps the boundary ∂Ω0 of the domain Ω0 homeomorphically onto the boundary ∂Ω of Ω and det ϕ (x) > 0 (x ∈ Ω0 ).

N satisfying boundary conditions x0 = 0, xN = 1 . It corresponds to the uniform ξ–grid ξi = i , i = 0, . . , N . N The mapping x(ξ) is approximated by xh : (ξ → x) constructed with the use of piecewise-linear finite element basis functions N xh (ξ) = xi ψi (ξi ) i=0 xi = xh (ξi ) . The Jacobian of the transformation xh (ξ) for ξ ∈ [i/N, (i+1)/N ] is equal to hi+1/2 dxh xi+1 − xi = = . dξ 1/N 1/N 32 Error estimates The Jacobian is positive if and only if hi+1/2 = xi+1 − xi > 0, i = 0, . . , N − 1 .

Let the coordinates of the unstructured grid be given (x, y)n , n = 1, . . , Nn . e. the array n(N, k) is also defined. 10) value summed over all mesh cell when using mapping of the unit square onto each cell of the mesh. It is clear that for a regular grid this formulation is reduced to a discrete analog of the problem of constructing harmonic coordinates ξ and η on the domain Ω. 10). 10) whereby the minimum ensures all mesh cells to be convex quadrilaterals and guarantees grid nondegeneracy.

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Bāǰalānī by MacKenzie, D.N.

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