# A posteriori error estimates for nonconforming approximation of elliptic problems based on Helmholtz type decomposition of the error

## Dr. Satyendra K. Tomar

**Jan. 22, 2008, 2:30 p.m. HF 136**

In this talk we shall present our work on a new type of a posteriori error estimates for nonconforming approximations of elliptic problems. The method is based on two key steps: (1) Helmholtz decomposition of vector-valued ﬁelds, and (2) splitting the residual functional with the help of an appropriate integral identity. We decompose the error into the ”gradient” and ”divergence-free” parts using the Helmholtz decomposition [1]. Following [3, 4] we then obtain computable two-sided bounds of the solutions of the resulting auxiliary problems. The estimates obtained differ from that was derived in [2, 5] using a projection to the conforming space. The numerical experiments conﬁrm that the a posteriori estimates derived with the help of Helmholtz decomposition are qualitatively same as those with projection arguments and can be efﬁciently exploited in practical computations.

Key words: Discontinuous Galerkin FEM; a posteriori error estimates; Helmholtz decomposition.

References

[1] Ainsworth, M.: A posteriori error estimation for discontinuous Galerkin ﬁnite element approximation. SIAM J. Numer. Anal. 45(4), 1777–1798 (2007).

[2] Lazarov, R., Repin, S., Tomar, S.: Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems. Numer. Methods Partial Differential Equations, accepted for publication.

[3] Repin, S.: A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp. 69(230), 481–500 (2000).

[4] Repin, S.: Two-sided estimates of deviation from exact solutions of uniformly elliptic equations. Proceedings of the St. Petersburg Mathematical Society, IX, 143–171, Amer. Math. Soc. Transl. Ser. 2, 209 (2003).

[5] Tomar, S., Repin, S.: Efﬁcient computable error bounds for discontinuous Galerkin approximations of elliptic problems. Submitted for publication, RICAM report 39 (2007).