Olaf Steinbach
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Olaf Steinbach has a total of 99 co-authors. Here are the top co-authors by number of shared publications:
- Günther Of
Graz University of Technology
25 shared publications - Ulrich Langer
Institute of Numerical Mathematics JKU Linz Austria
19 shared publications - Richard Löscher
Institut für Angewandte Mathematik Technische Universität Graz Steyrergasse 30 8010 Graz Austria
17 shared publications Wolfgang L. Wendland
University of Stuttgart
14 shared publications- Huidong Yang
Faculty of Mathematics Doppler Laboratory for Mathematical Mod- eling and Simulation of Next Generations of Ultrasound Devices (MaMSi) University of Vienna Oskar-Morgenstern- Platz 1 A-1090 Wien Austria
10 shared publications Huidong Yang
8 shared publications
Latest Publications
- 18 September 2026
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Numerical Analysis
- Vol. 64 (5) , 1982-2004
A Unified Framework for the Analysis, Numerical Approximation, and Model Reduction of Linear Operator Equations, Part I: Well-Posedness in Space and Time
We present a unified framework to construct well-posed formulations for large classes of linear operator equations including elliptic, parabolic, and hyperbolic partial differential equations. This general approach incorporates known weak variational formulations as well as novel space-time variational forms of the hyperbolic wave equation. The main concept is completion and extension of operators starting from the strong form of the problem. This paper lays the theoretical foundation for a unified approach toward numerical approximation methods and also model reduction of parameterized linear operator equations which will be the subject of the following parts.
- 18 September 2026
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Numerical Analysis
- Vol. 64 (5) , 1982-2004
A Unified Framework for the Analysis, Numerical Approximation, and Model Reduction of Linear Operator Equations, Part I: Well-Posedness in Space and Time
We present a unified framework to construct well-posed formulations for large classes of linear operator equations including elliptic, parabolic, and hyperbolic partial differential equations. This general approach incorporates known weak variational formulations as well as novel space-time variational forms of the hyperbolic wave equation. The main concept is completion and extension of operators starting from the strong form of the problem. This paper lays the theoretical foundation for a unified approach toward numerical approximation methods and also model reduction of parameterized linear operator equations which will be the subject of the following parts.
- 31 January 2026
- Published by Walter de Gruyter GmbH in Computational Methods in Applied Mathematics
Adaptive Least-Squares (Space-Time) Finite Element Methods For Convection-Diffusion Problems
In this paper we formulate and analyze adaptive (space-time) least-squares finite element methods for the solution of convection-diffusion equations. The convective derivative 𝒗 ⋅ ∇ u {\boldsymbol{v}\cdot\nabla u} is considered as part of the total time derivative d d t u = ∂ t u + 𝒗 ⋅ ∇ u {\frac{d}{dt}u=\partial_{t}u+\boldsymbol{v}\cdot\nabla u} , and therefore we can use a rather standard stability and error analysis for related space-time finite element methods. For stationary problems we restrict the ansatz space H 0 1 ( Ω ) {H^{1}_{0}(\Omega)} such that the convective derivative is considered as an element of the dual H - 1 ( Ω ) {H^{-1}(\Omega)} of the test space H 0 1 ( Ω ) {H^{1}_{0}(\Omega)} , which also allows unbounded velocities 𝒗 {\boldsymbol{v}} . While the discrete finite element schemes are always unique solvable, the numerical solutions may suffer from a bad approximation property of the finite element space when considering convection dominated problems, i.e., small diffusion coefficients. Instead of adding suitable stabilization terms, we aim to resolve the solutions by using adaptive (space-time) finite element methods. For this we introduce a least-squares approach where the discrete adjoint defines local a posteriori error indicators to drive an adaptive scheme. Numerical examples illustrate the theoretical considerations.
- 31 January 2026
- Published by Walter de Gruyter GmbH in Computational Methods in Applied Mathematics
Adaptive Least-Squares (Space-Time) Finite Element Methods For Convection-Diffusion Problems
In this paper we formulate and analyze adaptive (space-time) least-squares finite element methods for the solution of convection-diffusion equations. The convective derivative 𝒗 ⋅ ∇ u {\boldsymbol{v}\cdot\nabla u} is considered as part of the total time derivative d d t u = ∂ t u + 𝒗 ⋅ ∇ u {\frac{d}{dt}u=\partial_{t}u+\boldsymbol{v}\cdot\nabla u} , and therefore we can use a rather standard stability and error analysis for related space-time finite element methods. For stationary problems we restrict the ansatz space H 0 1 ( Ω ) {H^{1}_{0}(\Omega)} such that the convective derivative is considered as an element of the dual H - 1 ( Ω ) {H^{-1}(\Omega)} of the test space H 0 1 ( Ω ) {H^{1}_{0}(\Omega)} , which also allows unbounded velocities 𝒗 {\boldsymbol{v}} . While the discrete finite element schemes are always unique solvable, the numerical solutions may suffer from a bad approximation property of the finite element space when considering convection dominated problems, i.e., small diffusion coefficients. Instead of adding suitable stabilization terms, we aim to resolve the solutions by using adaptive (space-time) finite element methods. For this we introduce a least-squares approach where the discrete adjoint defines local a posteriori error indicators to drive an adaptive scheme. Numerical examples illustrate the theoretical considerations.
- Top Cited PapersOpen Access
- 27 January 2026
- Published by Springer Nature in Advances in Computational Mathematics
- Vol. 52 (1) , 7
Stable least-squares space-time boundary element methods for the wave equation
In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator $$\varvec{{\text {\textsf{V}}}}$$ for the wave equation as a minimization problem in $$\varvec{L^2(\Sigma )}$$ , where $$\varvec{\Sigma := \partial \Omega \times (0,T)}$$ is the lateral boundary of the space-time domain $$\varvec{Q:= \Omega \times (0,T)}$$ . For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature.- 27 January 2026
- Published by Springer Nature in Advances in Computational Mathematics
- Vol. 52 (1) , 7
Top Cited PapersOpen AccessStable least-squares space-time boundary element methods for the wave equation
In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator $$\varvec{{\text {\textsf{V}}}}$$ for the wave equation as a minimization problem in $$\varvec{L^2(\Sigma )}$$ , where $$\varvec{\Sigma := \partial \Omega \times (0,T)}$$ is the lateral boundary of the space-time domain $$\varvec{Q:= \Omega \times (0,T)}$$ . For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature. - 24 December 2025
- Published by Springer Nature in Numerische Mathematik
- Vol. 158 (2) , 653-669
An adaptive least squares boundary element method for elliptic boundary value problems
- 24 December 2025
- Published by Springer Nature in Numerische Mathematik
- Vol. 158 (2) , 653-669
An adaptive least squares boundary element method for elliptic boundary value problems
- 21 November 2025
- Published by Institute of Mathematics, Czech Academy of Sciences
- Vol. 70 (6) , 797-824
State-based approach to the numerical solution of Dirichlet boundary optimal control problems for the Laplace equation
Institute of Mathematics of the Czech Academy of Sciences
- 21 November 2025
- Published by Institute of Mathematics, Czech Academy of Sciences
- Vol. 70 (6) , 797-824
State-based approach to the numerical solution of Dirichlet boundary optimal control problems for the Laplace equation
Institute of Mathematics of the Czech Academy of Sciences
- Open Access
A unified framework for the analysis, numerical approximation and model reduction of linear operator equations, Part I: Well-posedness in space and time
We present a unified framework to construct well-posed formulations for large classes of linear operator equations including elliptic, parabolic and hyperbolic partial differential equations. This general approach incorporates known weak variational formulations as well as novel space-time variational forms of the hyperbolic wave equation. The main concept is completion and extension of operators starting from the strong form of the problem. This paper lays the theoretical foundation for a unified approach towards numerical approximation methods and also model reduction of parameterized linear operator equations which will be the subject of the following parts.
Open AccessA unified framework for the analysis, numerical approximation and model reduction of linear operator equations, Part I: Well-posedness in space and time
We present a unified framework to construct well-posed formulations for large classes of linear operator equations including elliptic, parabolic and hyperbolic partial differential equations. This general approach incorporates known weak variational formulations as well as novel space-time variational forms of the hyperbolic wave equation. The main concept is completion and extension of operators starting from the strong form of the problem. This paper lays the theoretical foundation for a unified approach towards numerical approximation methods and also model reduction of parameterized linear operator equations which will be the subject of the following parts.
- Open Access
State-based approach to the numerical solution of Dirichlet boundary optimal control problems for the Laplace equation
We investigate the Dirichlet boundary control of the Laplace equation, considering the control in $H^{1/2}(\partial Ω)$, which is the natural space for Dirichlet data when the state belongs to $H^1(Ω)$. The cost of the control is measured in the $H^{1/2}(\partial Ω)$ norm that also plays the role of the regularization term. We discuss regularization and finite element error estimates enabling us to derive an optimal relation between the finite element mesh size $h$ and the regularization parameter $\varrho$, balancing the energy cost for the control and the accuracy of the approximation of the desired state. This relationship is also crucial in designing efficient solvers. We also discuss additional box constraints imposed on the control and the state. Our theoretical findings are complemented by numerical examples, including one example with box constraints.Open AccessState-based approach to the numerical solution of Dirichlet boundary optimal control problems for the Laplace equation
We investigate the Dirichlet boundary control of the Laplace equation, considering the control in $H^{1/2}(\partial Ω)$, which is the natural space for Dirichlet data when the state belongs to $H^1(Ω)$. The cost of the control is measured in the $H^{1/2}(\partial Ω)$ norm that also plays the role of the regularization term. We discuss regularization and finite element error estimates enabling us to derive an optimal relation between the finite element mesh size $h$ and the regularization parameter $\varrho$, balancing the energy cost for the control and the accuracy of the approximation of the desired state. This relationship is also crucial in designing efficient solvers. We also discuss additional box constraints imposed on the control and the state. Our theoretical findings are complemented by numerical examples, including one example with box constraints. - Open Access
State-based nested iteration solution of optimal control problems with PDE constraints
We consider an abstract framework for the numerical solution of optimal control problems (OCPs) subject to partial differential equations (PDEs). Examples include not only the distributed control of elliptic PDEs such as the Poisson equation discussed in this paper in detail but also parabolic and hyperbolic equations. The approach covers the standard $L^2$ setting as well as the more recent energy regularization, also including state and control constraints. We discretize OCPs subject to parabolic or hyperbolic PDEs by means of space-time finite elements similar as in the elliptic case. We discuss regularization and finite element error estimates, and derive an optimal relation between the regularization parameter and the finite element mesh size in order to balance the accuracy, and the energy costs for the corresponding control. Finally, we also discuss the efficient solution of the resulting systems of algebraic equations, and their use in a state-based nested iteration procedure that allows us to compute finite element approximations to the state and the control in asymptotically optimal complexity. The numerical results illustrate the theoretical findings quantitatively.Open AccessState-based nested iteration solution of optimal control problems with PDE constraints
We consider an abstract framework for the numerical solution of optimal control problems (OCPs) subject to partial differential equations (PDEs). Examples include not only the distributed control of elliptic PDEs such as the Poisson equation discussed in this paper in detail but also parabolic and hyperbolic equations. The approach covers the standard $L^2$ setting as well as the more recent energy regularization, also including state and control constraints. We discretize OCPs subject to parabolic or hyperbolic PDEs by means of space-time finite elements similar as in the elliptic case. We discuss regularization and finite element error estimates, and derive an optimal relation between the regularization parameter and the finite element mesh size in order to balance the accuracy, and the energy costs for the corresponding control. Finally, we also discuss the efficient solution of the resulting systems of algebraic equations, and their use in a state-based nested iteration procedure that allows us to compute finite element approximations to the state and the control in asymptotically optimal complexity. The numerical results illustrate the theoretical findings quantitatively. - 28 April 2025
- Published by Springer Nature
- p. 283-291
A Hybrid Space-Time Finite Element Method For Parabolic Evolution Equations
- 28 April 2025
- Published by Springer Nature
- p. 283-291
A Hybrid Space-Time Finite Element Method For Parabolic Evolution Equations