Browsing by Author "Arnold, Douglas N."
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Item Analysis of a linear-linear finite element for the Reissner-Mindlin plate model(1996-03) Arnold, Douglas N.; Falk, R.S.Item Asymptotic estimates of hierarchical modeling(2003-10) Arnold, Douglas N.; Madureira, Alexandre L.In this paper we propose a way to analyze certain classes of dimension reduction models for elliptic problems in thin domains. We develop asymptotic expansions for the exact and model solutions, having the thickness as small parameter. The modeling error is then estimated by comparing the respective expansions, and the upper bounds obtained make clear the influence of the order of the model and the thickness on the convergence rates. The techniques developed here allows for estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers).Item Boundary conditions for the Einstein-Christoffel formulation of Einstein's equations(2005-10) Arnold, Douglas N.; Tarfulea, NicolaeItem Continuous Dependence on the Elastic Coefficients for a Class of Anisotropic Materials(1985) Arnold, Douglas N.; Falk, R.S.Item Differential complexes and numerical stability(2002-08) Arnold, Douglas N.Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial differential equations often hinges on capturing subtle aspects of the structure of the system in the discretization. In many cases the differential geometric structure captured by a differential complex has proven to be a key element, and a discrete differential complex which is appropriately related to the original complex is essential. This new geometric viewpoint has provided a unifying understanding of a variety of innovative numerical methods developed over recent decades and pointed the way to stable discretizations of problems for which none were previously known, and it appears likely to play an important role in attacking some currently intractable problems in numerical PDE.Item Differential complexes and stability of finite element methods. I. The de Rham complex(2005-02) Arnold, Douglas N.; Falk, Richard S.; Winther, RagnarItem Differential complexes and stability of finite element methods. II. The elasticity complex(2005-02) Arnold, Douglas N.; Falk, Richard S.; Winther, RagnarItem A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate(2003-10) Arnold, Douglas N.; Brezzi, Franco; Marini, L. DonatellaWe develop a family of locking-free elements for the Reissner-Mindlin plate using Discontinuous Galerkin techniques, one for each odd degree, and prove optimal error estimates. A second family uses conforming elements for the rotations and nonconforming elements for the transverse displacement, generalizing the element of Arnold and Falk to higher degree.Item Finite element exterior calculus, homological techniques, and applications(2006-02) Arnold, Douglas N.; Falk, Richard S.; Winther, RagnarItem Interior estimates for a low order finite element method for the Reissner-Mindlin plate model(1996-03) Arnold, Douglas N.; Liu, X.Item Locking-free Reissner-Mindlin elements without reduced integration(2005-12) Arnold, Douglas N.; Brezzi, Franco; Falk, Richard S.; Marini, L. DonatellaItem Mixed finite element methods for linear elasticity with weakly imposed symmetry(2005-10) Arnold, Douglas N.; Falk, Richard S.; Winther, RagnarItem New first-order formulation for the Einstein equations(2003-10) Alekseenko, Alexander M.; Arnold, Douglas N.We derive a new first-order formulation for Einstein's equations which involves fewer unknowns than other first-order formulations that have been proposed. The new formulation is based on the 3+1 decomposition with arbitrary lapse and shift. In the reduction to first order form only 8 particular combinations of the 18 first derivatives of the spatial metric are introduced. In the case of linearization about Minkowski space, the new formulation consists of symmetric hyperbolic system in 14 unknowns, namely the components of the extrinsic curvature perturbation and the 8 new variables, from whose solution the metric perturbation can be computed by integration.Item Nonconforming mixed elements for elasticity(2002-08) Arnold, Douglas N.; Winther, RagnarWe construct first order, stable, nonconforming mixed finite elements for plane elasticity and analyze their convergence. The mixed method is based on the Hellinger-Reissner variational formulation in which the stress and displacement fields are the primary unknowns. The stress elements use polynomial shape functions but do not involve vertex degrees of freedom.Item On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models(2002-08) Arnold, Douglas N.; Madureira, Alexandre L.; Zhang, ShengWe show that the Reissner-Mindlin plate bending model has a wider range of applicability than the Kirchhoff-Love model for the approximation of clamped linearly elastic plates. Under the assumption that the body force density is constant in the transverse direction, the Reissner-Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads. However, for loads with a significant transverse shear effect, the Kirchhoff-Love model fails.Item Preconditioning discrete approximations of the Reissner-Mindlin plate model(1996-05) Arnold, Douglas N.; Falk, R.S.; Winther, R.Item Preconditioning in H(div) and applications(1996-03) Arnold, Douglas N.; Falk, R.S.; Winther, R.Item Quadrilateral H(div) finite elements(2003-10) Arnold, Douglas N.; Boffi, Daniele; Falk, Richard S.We consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, extending results previously obtained for scalar approximation. The finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element, which is then transformed to a space of vector fields on each convex quadrilateral element via the Piola transform associated to a bilinear isomorphism of the square onto the element. For affine isomorphisms, a necessary and sufficientcondition for approximation of order r+1 in L2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms,the situation is more complicated and we give a precise characterization of what is needed for optimal order L2-approximation of the function and of its divergence. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of H(div). We also derive new estimates for approximation by quadrilateral Raviart-Thomas elements (requiring less regularity) and propose a new quadrilateral finite element space which provides optimal order approximation in H(div). Finally, we demonstrate the theory with numerical computations of mixed and least squares finite element aproximations of the solution of Poisson's equation.Item Rectangular mixed finite elements for elasticity(2005-02) Arnold, Douglas N.; Awanou, GerardItem Regular Inversion of the Divergence Operator with Dirichlet Boundary Conditions on a Polygon.(1987) Arnold, Douglas N.; Scott, L. Ridgway; Vogelius, M.