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A twolevel additive Schwarz preconditioner for nonconforming plate elements
 Numer. Math
, 1994
"... Abstract. Twolevel additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar secondorder symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergencefree no ..."
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Cited by 58 (5 self)
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Abstract. Twolevel additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar secondorder symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergencefree nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap. 1.
A family of higher order mixed finite element methods for plane elasticity
 Numer. Math
, 1984
"... Summary. The Dirichlet problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the RaviartThomas mixed finite elements for a scalar, secondorder elliptic equatio ..."
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Cited by 49 (11 self)
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Summary. The Dirichlet problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the RaviartThomas mixed finite elements for a scalar, secondorder elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor in L2(f2), and optimal order negative norm estimates are obtained in H=(g2) ' for a range of s depending on the index of the finite element space. An optimal order estimate in L~176 for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail. Subject Classifications: AMS(MOS): 65N30, CR: G 1.8. 1.
Convergence of nonconforming multigrid methods without full elliptic regularity
 Math. Comp
, 1995
"... Abstract. We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (< 1) for the contraction number of the Wcycle algorithm which is independent ..."
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Cited by 23 (0 self)
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Abstract. We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (< 1) for the contraction number of the Wcycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large. We also show that the symmetric variable Vcycle algorithm is an optimal preconditioner. 1.
C¹ ConvexityPreserving Interpolation of Scattered Data
 SIAM J. COMPUT. VOL. 20, NO. 5, PP. 17321752
, 1999
"... We describe a detailed computational procedure which, given data values at arbitrarily distributed points in the plane, determines if the data are convex and, if so, constructs a smooth convex surface that interpolates the data. The method consists of constructing a triangulation of the nodes (data ..."
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Cited by 3 (0 self)
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We describe a detailed computational procedure which, given data values at arbitrarily distributed points in the plane, determines if the data are convex and, if so, constructs a smooth convex surface that interpolates the data. The method consists of constructing a triangulation of the nodes (data abscissae) for which the trianglebased piecewise linear interpolant is convex, computing a set of nodal gradients for which there exists a convex Hermite interpolant, and constructing a smooth convex surface that interpolates the nodal values and gradients. The method involves two datadependent triangulations along with a straightline dual of each, and we describe some interesting relationships among them.
On the Full C1Qk Finite Element Spaces on Rectangles and Cuboids
, 2010
"... Abstract. We study the extensions of the BognerFoxSchmit element to the whole family of Qk continuously differentiable finite elements on rectangular grids, for all k≥3, in 2D and 3D. We show that the newly defined C1 spaces are maximal in the sense that they contain all C1Qk functions of piecewi ..."
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Cited by 2 (1 self)
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Abstract. We study the extensions of the BognerFoxSchmit element to the whole family of Qk continuously differentiable finite elements on rectangular grids, for all k≥3, in 2D and 3D. We show that the newly defined C1 spaces are maximal in the sense that they contain all C1Qk functions of piecewise polynomials. We give examples of other extensions of C1Qk elements. The result is consistent with the Strang’s conjecture (restricted to the quadrilateral grids in 2D and 3D). Some numerical results are provided on the family of C1 elements solving the biharmonic equation. AMS subject classifications: 65M60, 65N30
Implicit and FourthOrder Accurate SemiLagrangian Contouring for Geometric Moving Interface Problems by
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Thin Film and NonNewtonian Flow Problems
"... This research deals with several novel aspects of finite element formulations and methodology in parallel adaptive simulation of flow problems. Composite macroelement schemes are developed for problems of thin fluid layers with deforming free surfaces or decomposing material phases; experiments are ..."
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This research deals with several novel aspects of finite element formulations and methodology in parallel adaptive simulation of flow problems. Composite macroelement schemes are developed for problems of thin fluid layers with deforming free surfaces or decomposing material phases; experiments are also run on divergencefree formulations that can be derived from the same element classes. The constrained composite nature and C1 continuity requirements of these elements raises new issues, especially with respect to adaptive refinement patterns and the treatment of hanging node constraints, which are more complex than encountered with standard element types. This work combines such complex elements with these applications and with parallel adaptive mesh refinement and coarsening (AMR/C) techniques for the first time. The use of adaptive macroelement spaces also requires appropriate programming interfaces and data structures to enable easy and efficient implementation in parallel software. The algorithms developed for this work are implemented using objectoriented designs described herein.
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"... family of continuously differentiable finite elements on simplicial grids in four space dimensions Shangyou Zhang∗ A family of continuously differentiable piecewise polynomials of degree k, for all k ≥ 17, on general 4D simplicial grids, is constructed. Such a finite element space assumes full order ..."
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family of continuously differentiable finite elements on simplicial grids in four space dimensions Shangyou Zhang∗ A family of continuously differentiable piecewise polynomials of degree k, for all k ≥ 17, on general 4D simplicial grids, is constructed. Such a finite element space assumes full order of approximation. As a byproduct, we obtain a family of special 3D C2Pk elements on tetrahedral grids.