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 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 Preconditioning discrete approximations of the Reissner-Mindlin plate model(1996-05) 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 Unified analysis of discontinuous Galerkin methods for elliptic problems(2001-02) Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo; Marini, L. DonatellaWe provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems by diverse communities over three decades.Item A Uniformly Accurate Finite Element Method for Mindlin-Reissner Plate(1987) Arnold, Douglas N.; Falk, R.S.