<p>Least-squares discretization schemes<br/>【2026.08.17 10:00-11:00，N202】</p>----中国科学院国家数学与交叉科学中心 (NCMIS)

2026.07.27

Colloquia Seminars

Speaker	Prof. Olaf Steinbach, Institute of Applied Mathematics at Graz University of Technology
Title	Least-squares discretization schemes
Time	2026.08.17 10:00-11:00
Venue	N202
Abstract	In this talk, we discuss the least-squares discretization of operator equations Bu=f where B:X→Y^is assumed to be invertible when considering suitable Hilbert spaces Xand Y. Choosing a self-adjoint and elliptic operatorA:Y→Y^ we consider the minimization of 1/2\|\|Bu−f\|\|_A⁻¹², where the minimizer u=B⁻¹f solves the gradient equation B^A⁻¹Bu=B^A⁻¹f. When introducing the adjoint variable p=A⁻¹(f−Bu), we conclude the systemAp+Bu=f,B^*p=0 to be solved by some mixed discretization scheme. While for the solution (u,p)∈X×Y we conclude Bu=f and p≡0, we can use p_h∈Y_h as a posteriori error indicator to drive adaptive schemes. The abstract theory is given in our recent publication [2], where we discuss the elliptic diffusion equation, the parabolic heat equation, and the hyperbolic wave equation. In addition, [3] covers the case of the advection-diffusion equation. A more prominent example is the pressure recovery in fluid mechanics, when the velocity is known [4]. Other applications involve boundary element methods in the case of less regular data [5], and least-squares space-time boundary element methods for the wave equation [1]. References [1] D. Hoonhout, R. Löscher, O. Steinbach, C. Urzua-Torres: Stable least-squares space-time boundary element methods for the wave equation.Adv. Comput. Math.52, 7 (2026). [2] C. Köthe, R. Löscher, O. Steinbach: Adaptive least-squares space-time finite element methods.Electron. Trans. Numer. Anal., to appear, 2026. [3] C. Köthe, O. Steinbach: Adaptive least-squares space-time finite element methods for convection-diffusion problems.Comput. Methods Appl. Math.26(2026) 211-239. [4] D. R. Q. Pacheco, O. Steinbach: Optimal pressure recovery using an ultra-weak finite element method for the pressure Poisson equation and a least-squares approach for the gradient equation.Comput. Methods Appl. Math.24(2024) 921-934. [5] O. Steinbach: An adaptive least squares boundary element method for elliptic boundary value problems.Numer. Math.158(2026) 653-669.
Biography	Prof. Dr. Olaf Steinbach is the Head of the Institute of Applied Mathematics at Graz University of Technology (TU Graz), Austria. He obtained his diploma in Mathematics from the Technical University of Chemnitz in 1992 and received his doctorate from the University of Stuttgart in 1996, where he also completed his habilitation in 2001. After visiting professorships at TU Chemnitz, Johannes Kepler University Linz, and TU Dresden, he was appointed to his current position at TU Graz in 2004. He is a founding member of the Graz Center for Computational Engineering. Prof. Steinbach’s research focuses on the numerical analysis and fast solution of partial differential equations. His main areas include boundary element methods (BEM), finite element methods (FEM), and their coupling; domain decomposition techniques; space‐time finite element methods for time‐dependent problems; and optimal control of PDEs with energy regularization. He has authored or co‐authored over 100 journal articles many of those have appeared in leading journals, 7 monographs, and numerous book chapters and proceedings papers.

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