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We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided a priori bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis....
We derive an optimal lower bound of the
interpolation error for linear finite elements on a bounded two-dimensional
domain. Using the supercloseness between the linear interpolant
of the true solution of an elliptic problem and its finite element
solution on uniform partitions, we further
obtain two-sided a priori bounds of the discretization error by means of the
interpolation error. Two-sided bounds for bilinear finite elements
are given as well. Numerical tests illustrate our theoretical
analysis.
...
In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong weight with respect to the metric associated with the operator. Roughly speaking, the strong condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations....
We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by . We associate to Problem an optimal control problem, denoted by . Then, using appropriate Tykhonov triples, governed by a nonlinear operator and a convex , we provide results concerning the well-posedness of problems...
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