# On the stability of the L2 projection and the quasiinterpolant in the space of smooth periodic splines

@article{Dougalis2021OnTS, title={On the stability of the L2 projection and the quasiinterpolant in the space of smooth periodic splines}, author={D. A. Dougalis}, journal={ArXiv}, year={2021}, volume={abs/2106.09060} }

In this paper we derive stability estimates in Land L∞based Sobolev spaces for the L projection and a family of quasiinterolants in the space of smooth, 1-periodic, polynomial splines defined on a uniform mesh in [0, 1]. As a result of the assumed periodicity and the uniform mesh, cyclic matrix techniques and suitable decay estimates of the elements of the inverse of a Gram matrix associated with the standard basis of the space of splines, are used to establish the stability results.

#### One Citation

On the well-posedness of the Galerkin semidiscretization of the periodic initial-value problem of the Serre equations

- Computer Science, Mathematics
- ArXiv
- 2021

It is proved that the semidiscrete problem is well posed, locally in time, and satisfies a discrete positivity property for the water depth. Expand

#### References

SHOWING 1-10 OF 16 REFERENCES

Convergence Estimates for Galerkin Methods for Variable Coefficient Initial Value Problems

- Mathematics
- 1974

The use of Galerkin’s method for the approximate solution of the initial value problem for certain simple equations ${{\partial u} / {\partial t = Pu}}$, where P is a differential operator of order m… Expand

On the standard Galerkin method with explicit RK4 time stepping for the Shallow Water equations

- Mathematics
- 2018

We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension. We discretize the problem in space by the standard Galerkin finite element method on a… Expand

Error estimates for the standard Galerkin-finite element method for the shallow water equations

- Mathematics, Computer Science
- Math. Comput.
- 2016

It is shown that in the case of spatial discretizations with piecewise linear continuous functions on a uniform mesh, suitable compatibility conditions at the boundary and superaccuracy properties of the L2 projection on the finite element subspaces lead to an optimal-order O(h2) L2-error estimate. Expand

GALERKIN APPROXIMATIONS OF PERIODIC SOLUTIONS OF BOUSSINESQ SYSTEMS

- Mathematics
- 2010

We consider the periodic initial-value problem for the family of a-b-c-d Boussinesq systems, [8], [9], and their completely symmetric analogs, [10]. We approximate their solutions by the standard… Expand

Optimal Lœ Error Estimates for Galerkin Approximations to Solutions of Two-Point Boundary Value Problems

- 2010

A priori error estimates in the maximum norm are derived for Galerkin approximations to solutions of two-point boundary valué problems. The class of Galerkin spaces considered includes almost all… Expand

The Stability in- L and W^ of the L2-Projection onto Finite Element Function Spaces

- 2010

The stability of the Z.2-projection onto some standard finite element spaces Vh, considered as a map in Lp and W^, 1 ^ p < oo, is shown under weaker regularity requirements than quasi-uniformity of… Expand

Conservative

- high-order numerical schemes for the generalized Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London, A351
- 1995

Conservative, high-order numerical schemes for the generalized Korteweg—de Vries equation

- Mathematics
- Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences
- 1995

A class of fully discrete schemes for the numerical simulation of solutions of the periodic initial-value problem for a class of generalized Korteweg-de Vries equations is analysed, implemented and… Expand

On some high order accurate fully discrete methods for the Korteweg-de Vries equation

- Math. Comp., 45
- 1985

Decay rates for inverses of band matrices

- Mathematics
- 1984

Spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices. The rate of decay of A 1 can be bounded in… Expand