36 research outputs found
A posteriori error estimates for nonconforming discretizations of singularly perturbed biharmonic operators
For the pure biharmonic equation and a biharmonic singular perturbation
problem, a residual-based error estimator is introduced which applies to many
existing nonconforming finite elements. The error estimator involves the local
best-approximation error of the finite element function by piecewise polynomial
functions of the degree determining the expected approximation order, which
need not coincide with the maximal polynomial degree of the element, for
example if bubble functions are used. The error estimator is shown to be
reliable and locally efficient up to this polynomial best-approximation error
and oscillations of the right-hand side
Two families of -rectangle nonconforming finite elements for sixth-order elliptic equations
In this paper, we propose two families of nonconforming finite elements on
-rectangle meshes of any dimension to solve the sixth-order elliptic
equations. The unisolvent property and the approximation ability of the new
finite element spaces are established. A new mechanism, called the exchange of
sub-rectangles, for investigating the weak continuities of the proposed
elements is discovered. With the help of some conforming relatives for the
problems, we establish the quasi-optimal error estimate for the
tri-harmonic equation in the broken norm of any dimension. The
theoretical results are validated further by the numerical tests in both 2D and
3D situations
A family of stabilizer-free virtual elements on triangular meshes
A family of stabilizer-free virtual elements are constructed on
triangular meshes. When choosing an accurate and proper interpolation, the
stabilizer of the virtual elements can be dropped while the quasi-optimality is
kept. The interpolating space here is the space of continuous polynomials
on the Hsieh-Clough-Tocher macro-triangle, where the macro-triangle is defined
by connecting three vertices of a triangle with its barycenter. We show that
such an interpolation preserves polynomials locally and enforces the
coerciveness of the resulting bilinear form. Consequently the stabilizer-free
virtual element solutions converge at the optimal order. Numerical tests are
provided to confirm the theory and to be compared with existing virtual
elements