Optimal error estimates for corrected trapezoidal rules

-Stable Derivative-Free Error-Corrected Trapezoidal Rule for Burgers' Equation with Inconsistent Initial and Boundary Conditions

Equivalent a posteriori error estimates for spectral element solutions of constrained optimal control problem in one dimension

‎In this paper‎, ‎we study spectral element approximation for a constrained‎ ‎optimal control problem in one dimension‎. ‎The equivalent a posteriori error estimators are derived for‎ ‎the control‎, ‎the state and the adjoint state approximation‎. ‎Such estimators can be used to‎ ‎construct adaptive spectral elements for the control problems.

Hybrid Gauss-Trapezoidal Quadrature Rules

A new class of quadrature rules for the integration of both regular and singular functions is constructed and analyzed. For each rule the quadrature weights are positive and the class includes rules of arbitrarily high-order convergence. The quadratures result from alterations to the trapezoidal rule, in which a small number of nodes and weights at the ends of the integration interval are repla...

Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations

It is shown that the Ritz projection onto spaces of piecewise linear finite elements is bounded in the Sobolev space, Wp\ for 2 <p < oo. This implies that for functions in W¿ n Wp the error in approximation behaves like 0(h) in Wx, for 2 <p =c oo, and like 0(h2) in Lp, for 2 *íp < oo. In all these cases the additional logarithmic factor previously included in error estimates for linear finite e...

Exact Error Estimates and Optimal Randomized Algorithms for Integration

Exact error estimates for evaluating multi-dimensional integrals are considered. An estimate is called exact if the rates of convergence for the lowand upper-bound estimate coincide. The algorithm with such an exact rate is called optimal. Such an algorithm has an unimprovable rate of convergence. The problem of existing exact estimates and optimal algorithms is discussed for some functional sp...