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Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra

Gunar Matthies (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

We present families of scalar nonconforming finite elements of arbitrary order r 1 with optimal approximation properties on quadrilaterals and hexahedra. Their vector-valued versions together with a discontinuous pressure approximation of order r - 1 form inf-sup stable finite element pairs of order r for the Stokes problem. The well-known elements by Rannacher and Turek are recovered in the case r=1. A numerical comparison between conforming and nonconforming discretisations will be given. Since higher order...

Inner products in covolume and mimetic methods

Kathryn A. Trapp (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This...

Integrability for very weak solutions to boundary value problems of p -harmonic equation

Hongya Gao, Shuang Liang, Yi Cui (2016)

Czechoslovak Mathematical Journal

The paper deals with very weak solutions u θ + W 0 1 , r ( Ω ) , max { 1 , p - 1 } < r < p < n , to boundary value problems of the p -harmonic equation - div ( | u ( x ) | p - 2 u ( x ) ) = 0 , x Ω , u ( x ) = θ ( x ) , x Ω . ( * ) We show that, under the assumption θ W 1 , q ( Ω ) , q > r , any very weak solution u to the boundary value problem ( * ) is integrable with u θ + L weak q * ( Ω ) for q < n , θ + L weak τ ( Ω ) for q = n and any τ < , θ + L ( Ω ) for q > n , provided that r is sufficiently close to p .

Integral inequalities and summability of solutions of some differential problems

Lucio Boccardo (2000)

Banach Center Publications

The aim of this note is to indicate how inequalities concerning the integral of | u | 2 on the subsets where |u(x)| is greater than k ( k I R + ) can be used in order to prove summability properties of u (joint work with Daniela Giachetti). This method was introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems. In some joint works with Thierry Gallouet, inequalities concerning the integral of | u | 2 on the subsets where |u(x)| is less than k ( k I R + ) or...

Internal finite element approximations in the dual variational method for second order elliptic problems with curved boundaries

Ivan Hlaváček, Michal Křížek (1984)

Aplikace matematiky

Using the stream function, some finite element subspaces of divergence-free vector functions, the normal components of which vanish on a part of the piecewise smooth boundary, are constructed. Applying these subspaces, an internal approximation of the dual problem for second order elliptic equations is defined. A convergence of this method is proved without any assumption of a regularity of the solution. For sufficiently smooth solutions an optimal rate of convergence is proved. The internal approximation...

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