Browsing by Author "Falk, Richard S."
Now showing 1 - 6 of 6
Results Per Page
Sort Options
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 Finite element exterior calculus, homological techniques, and applications(2006-02) Arnold, Douglas N.; Falk, Richard S.; Winther, RagnarItem 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 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.