Joachim Schöberl - Academia.edu (original) (raw)

Papers by Joachim Schöberl

arXiv (Cornell University), Nov 30, 2021

We introduce two new lowest order methods, a mixed method, and a hybrid Discontinuous Galerkin (H... more We introduce two new lowest order methods, a mixed method, and a hybrid Discontinuous Galerkin (HDG) method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi-Douglas-Marini space for approximating the velocity and the lowest order Raviart-Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

Journal of Scientific Computing

We introduce two new lowest order methods, a mixed method, and a hybrid discontinuous Galerkin me... more We introduce two new lowest order methods, a mixed method, and a hybrid discontinuous Galerkin method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi–Douglas–Marini space for approximating the velocity and the lowest order Raviart–Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

arXiv (Cornell University), Jan 6, 2012

A uniform inf-sup condition related to a parameter dependent Stokes problem is established. Such ... more A uniform inf-sup condition related to a parameter dependent Stokes problem is established. Such conditions are intimately connected to the construction of uniform preconditioners for the problem, i.e., preconditioners which behave uniformly well with respect to variations in the model parameter as well as the discretization parameter. For the present model, similar results have been derived before, but only by utilizing extra regularity ensured by convexity of the domain. The purpose of this paper is to remove this artificial assumption. As a byproduct of our analysis, in the two dimensional case we also construct a new projection operator for the Taylor-Hood element which is uniformly bounded in L 2 and commutes with the divergence operator. This construction is based on a tight connection between a subspace of the Taylor-Hood velocity space and the lowest order Nedelec edge element.

arXiv (Cornell University), Jun 19, 2022

The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and su... more The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that certain Regge approximations yield curvature and connection approximations that converge at a higher rate than previously known. The analysis is based on covariant (distributional) curl and incompatibility operators which can be applied to piecewise smooth matrix fields whose tangential-tangential component is continuous across element interfaces. Using the properties of the canonical interpolant of the Regge space, we obtain superconvergence of approximations of these covariant operators. Numerical experiments further illustrate the results from the error analysis.

Recent works showed that pressure-robust modifications of mixed finite element methods for the St... more Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure-independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the L 2-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier-Stokes equations. The novel error estimators only take the ...

Lecture Notes in Computational Science and Engineering, 2020

We present a new numerical method for solving time dependent Maxwell equations, which is also sui... more We present a new numerical method for solving time dependent Maxwell equations, which is also suitable for general linear hyperbolic equations. It is based on an unstructured partitioning of the spacetime domain into tent-shaped regions that respect causality. Provided that an approximate solution is available at the tent bottom, the equation can be locally evolved up to the top of the tent. By mapping tents to a domain which is a tensor product of a spatial domain with a time interval, it is possible to construct a fully explicit scheme that advances the solution through unstructured meshes. This work highlights a difficulty that arises when standard explicit Runge Kutta schemes are used in this context and proposes an alternative structure-aware Taylor time-stepping technique. Thus explicit methods are constructed that allow variable time steps and local refinements without compromising high order accuracy in space and time. These Mapped Tent Pitching (MTP) schemes lead to highly ...

In this article, we consider a hybridized tangential-displacement normal-normalstress (TDNNS) dis... more In this article, we consider a hybridized tangential-displacement normal-normalstress (TDNNS) discretization of linear elasticity. As shown in earlier work by Joachim Schöberl and the first author, TDNNS is a stable finite element discretization that does not suffers from volume locking. We propose a finite element tearing and interconnecting (FETI) method in order to solve the resulting linear system iteratively. The method is analyzed thoroughly for the compressible case in two dimensions, leading to a condition number bound of C(1 + log(H/h))2, which coincides with known bounds of many other iterative substructuring methods. Numerical results confirm our theoretical findings. Furthermore, our experiments show that a certain instance of the method remains stable even in the almost incompressible limit.

Computer Methods in Applied Mechanics and Engineering, 2021

SIAM Journal on Scientific Computing, 2017

A spacetime domain can be progressively meshed by tent shaped objects. Numerical methods for solv... more A spacetime domain can be progressively meshed by tent shaped objects. Numerical methods for solving hyperbolic systems using such tent meshes to advance in time have been proposed previously. Such schemes have the ability to advance in time by different amounts at different spatial locations. This paper explores a technique by which standard discretizations, including explicit time stepping, can be used within tentshaped spacetime domains. The technique transforms the equations within a spacetime tent to a domain where space and time are separable. After detailing techniques based on this mapping, several examples including the acoustic wave equation and the Euler system are considered.

Numerische Mathematik, 2019

Recent works showed that pressure-robust modifications of mixed finite element methods for the St... more Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the L 2-norm of the right-hand side. However, the velocity is only steered by the divergencefree part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier-Stokes equations. The novel error estimators only take the curl of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressuredominant situations. This is also confirmed by some numerical examples with the novel pressurerobust modifications of the Taylor-Hood and mini finite element methods. incompressible Navier-Stokes equations and mixed finite elements and pressure robustness and a posteriori error estimators and adaptive mesh refinement

Computers & Mathematics with Applications, 2015

In this paper we present a Discontinuous Galerkin method for the Boltzmann equation. The distribu... more In this paper we present a Discontinuous Galerkin method for the Boltzmann equation. The distribution function f is approximated by a shifted Maxwellian times a polynomial in space and momentum, while the test functions are chosen as polynomials. The first property leads to consistency with the Euler limit, while the second property ensures conservation of mass, momentum and energy. The focus of the paper is on efficient algorithms for the Boltzmann collision operator. We transform between nodal, hierarchical and polar polynomial bases to reduce the inner integral operator to diagonal form.

IFAC Proceedings Volumes, 2012

SIAM Journal on Scientific Computing, 2009

SIAM Journal on Numerical Analysis, 2011

We consider the discretized optimality system of a special class of elliptic optimal control prob... more We consider the discretized optimality system of a special class of elliptic optimal control problems and propose an all-at-once multigrid method for solving this discretized system. Under standard assumptions the convergence of the multigrid method and the robustness of the convergence rates with respect to the involved parameter are shown. Numerical experiments are presented for illustrating the theoretical results.

SIAM Journal on Numerical Analysis, 2008

Numerische Mathematik, 2003

We consider an interior penalty discontinuous approximation for symmetric elliptic problems of se... more We consider an interior penalty discontinuous approximation for symmetric elliptic problems of second order on non-matching grids in this paer. The main result is an almost optimal error estimate for the interior penalty approximation of the original problem based on the partition of the domain into a finite number of subdomains. Further, an error analysis for the finite element approximation of the penalty formulation is given. Finally, numerical experiments on a series of model second order problems are presented.

Mathematics of Computation, 2012

Mathematics of Computation, 2007

Inverse Problems, 2010

In this paper we investigate the efficient realization of sampling methods based on solutions of ... more In this paper we investigate the efficient realization of sampling methods based on solutions of certain adjoint problems. This adjoint approach does not require the explicit knowledge of the Green's function for the background medium, and allows us to sample for all points and all dipole directions simultaneously; thus, several limitations of standard sampling methods are relieved. A detailed derivation of the adjoint approach is presented for two electromagnetic model problems, but the framework can be applied to a much wider class of problems. We also discuss a relation of the adjoint sampling method to standard backprojection algorithms, and present numerical tests that illustrate the efficiency of the adjoint approach.

International Journal for Numerical Methods in Engineering, 2011

SUMMARYIn this paper, we present a family of mixed finite elements, which are suitable for the di... more SUMMARYIn this paper, we present a family of mixed finite elements, which are suitable for the discretization of slim domains. The displacement space is chosen as Nédélec's space of tangential continuous elements, whereas the stress is approximated by normal–normal continuous symmetric tensor‐valued finite elements. We show stability of the system on a slim domain discretized by a tensor product mesh, where the constant of stability does not depend on the aspect ratio of the discretization. We give interpolation operators for the finite element spaces, and thereby obtain optimal order a priori error estimates for the approximate solution. All estimates are independent of the aspect ratio of the finite elements. Copyright © 2011 John Wiley & Sons, Ltd.

arXiv (Cornell University), Nov 30, 2021

We introduce two new lowest order methods, a mixed method, and a hybrid Discontinuous Galerkin (H... more We introduce two new lowest order methods, a mixed method, and a hybrid Discontinuous Galerkin (HDG) method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi-Douglas-Marini space for approximating the velocity and the lowest order Raviart-Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

Journal of Scientific Computing

We introduce two new lowest order methods, a mixed method, and a hybrid discontinuous Galerkin me... more We introduce two new lowest order methods, a mixed method, and a hybrid discontinuous Galerkin method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi–Douglas–Marini space for approximating the velocity and the lowest order Raviart–Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

arXiv (Cornell University), Jan 6, 2012

A uniform inf-sup condition related to a parameter dependent Stokes problem is established. Such ... more A uniform inf-sup condition related to a parameter dependent Stokes problem is established. Such conditions are intimately connected to the construction of uniform preconditioners for the problem, i.e., preconditioners which behave uniformly well with respect to variations in the model parameter as well as the discretization parameter. For the present model, similar results have been derived before, but only by utilizing extra regularity ensured by convexity of the domain. The purpose of this paper is to remove this artificial assumption. As a byproduct of our analysis, in the two dimensional case we also construct a new projection operator for the Taylor-Hood element which is uniformly bounded in L 2 and commutes with the divergence operator. This construction is based on a tight connection between a subspace of the Taylor-Hood velocity space and the lowest order Nedelec edge element.

arXiv (Cornell University), Jun 19, 2022

The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and su... more The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that certain Regge approximations yield curvature and connection approximations that converge at a higher rate than previously known. The analysis is based on covariant (distributional) curl and incompatibility operators which can be applied to piecewise smooth matrix fields whose tangential-tangential component is continuous across element interfaces. Using the properties of the canonical interpolant of the Regge space, we obtain superconvergence of approximations of these covariant operators. Numerical experiments further illustrate the results from the error analysis.

Recent works showed that pressure-robust modifications of mixed finite element methods for the St... more Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure-independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the L 2-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier-Stokes equations. The novel error estimators only take the ...

Lecture Notes in Computational Science and Engineering, 2020

We present a new numerical method for solving time dependent Maxwell equations, which is also sui... more We present a new numerical method for solving time dependent Maxwell equations, which is also suitable for general linear hyperbolic equations. It is based on an unstructured partitioning of the spacetime domain into tent-shaped regions that respect causality. Provided that an approximate solution is available at the tent bottom, the equation can be locally evolved up to the top of the tent. By mapping tents to a domain which is a tensor product of a spatial domain with a time interval, it is possible to construct a fully explicit scheme that advances the solution through unstructured meshes. This work highlights a difficulty that arises when standard explicit Runge Kutta schemes are used in this context and proposes an alternative structure-aware Taylor time-stepping technique. Thus explicit methods are constructed that allow variable time steps and local refinements without compromising high order accuracy in space and time. These Mapped Tent Pitching (MTP) schemes lead to highly ...

In this article, we consider a hybridized tangential-displacement normal-normalstress (TDNNS) dis... more In this article, we consider a hybridized tangential-displacement normal-normalstress (TDNNS) discretization of linear elasticity. As shown in earlier work by Joachim Schöberl and the first author, TDNNS is a stable finite element discretization that does not suffers from volume locking. We propose a finite element tearing and interconnecting (FETI) method in order to solve the resulting linear system iteratively. The method is analyzed thoroughly for the compressible case in two dimensions, leading to a condition number bound of C(1 + log(H/h))2, which coincides with known bounds of many other iterative substructuring methods. Numerical results confirm our theoretical findings. Furthermore, our experiments show that a certain instance of the method remains stable even in the almost incompressible limit.

Computer Methods in Applied Mechanics and Engineering, 2021

SIAM Journal on Scientific Computing, 2017

A spacetime domain can be progressively meshed by tent shaped objects. Numerical methods for solv... more A spacetime domain can be progressively meshed by tent shaped objects. Numerical methods for solving hyperbolic systems using such tent meshes to advance in time have been proposed previously. Such schemes have the ability to advance in time by different amounts at different spatial locations. This paper explores a technique by which standard discretizations, including explicit time stepping, can be used within tentshaped spacetime domains. The technique transforms the equations within a spacetime tent to a domain where space and time are separable. After detailing techniques based on this mapping, several examples including the acoustic wave equation and the Euler system are considered.

Numerische Mathematik, 2019

Recent works showed that pressure-robust modifications of mixed finite element methods for the St... more Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the L 2-norm of the right-hand side. However, the velocity is only steered by the divergencefree part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier-Stokes equations. The novel error estimators only take the curl of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressuredominant situations. This is also confirmed by some numerical examples with the novel pressurerobust modifications of the Taylor-Hood and mini finite element methods. incompressible Navier-Stokes equations and mixed finite elements and pressure robustness and a posteriori error estimators and adaptive mesh refinement

Computers & Mathematics with Applications, 2015

In this paper we present a Discontinuous Galerkin method for the Boltzmann equation. The distribu... more In this paper we present a Discontinuous Galerkin method for the Boltzmann equation. The distribution function f is approximated by a shifted Maxwellian times a polynomial in space and momentum, while the test functions are chosen as polynomials. The first property leads to consistency with the Euler limit, while the second property ensures conservation of mass, momentum and energy. The focus of the paper is on efficient algorithms for the Boltzmann collision operator. We transform between nodal, hierarchical and polar polynomial bases to reduce the inner integral operator to diagonal form.

IFAC Proceedings Volumes, 2012

SIAM Journal on Scientific Computing, 2009

SIAM Journal on Numerical Analysis, 2011

We consider the discretized optimality system of a special class of elliptic optimal control prob... more We consider the discretized optimality system of a special class of elliptic optimal control problems and propose an all-at-once multigrid method for solving this discretized system. Under standard assumptions the convergence of the multigrid method and the robustness of the convergence rates with respect to the involved parameter are shown. Numerical experiments are presented for illustrating the theoretical results.

SIAM Journal on Numerical Analysis, 2008

Numerische Mathematik, 2003

We consider an interior penalty discontinuous approximation for symmetric elliptic problems of se... more We consider an interior penalty discontinuous approximation for symmetric elliptic problems of second order on non-matching grids in this paer. The main result is an almost optimal error estimate for the interior penalty approximation of the original problem based on the partition of the domain into a finite number of subdomains. Further, an error analysis for the finite element approximation of the penalty formulation is given. Finally, numerical experiments on a series of model second order problems are presented.

Mathematics of Computation, 2012

Mathematics of Computation, 2007

Inverse Problems, 2010

In this paper we investigate the efficient realization of sampling methods based on solutions of ... more In this paper we investigate the efficient realization of sampling methods based on solutions of certain adjoint problems. This adjoint approach does not require the explicit knowledge of the Green's function for the background medium, and allows us to sample for all points and all dipole directions simultaneously; thus, several limitations of standard sampling methods are relieved. A detailed derivation of the adjoint approach is presented for two electromagnetic model problems, but the framework can be applied to a much wider class of problems. We also discuss a relation of the adjoint sampling method to standard backprojection algorithms, and present numerical tests that illustrate the efficiency of the adjoint approach.

International Journal for Numerical Methods in Engineering, 2011

SUMMARYIn this paper, we present a family of mixed finite elements, which are suitable for the di... more SUMMARYIn this paper, we present a family of mixed finite elements, which are suitable for the discretization of slim domains. The displacement space is chosen as Nédélec's space of tangential continuous elements, whereas the stress is approximated by normal–normal continuous symmetric tensor‐valued finite elements. We show stability of the system on a slim domain discretized by a tensor product mesh, where the constant of stability does not depend on the aspect ratio of the discretization. We give interpolation operators for the finite element spaces, and thereby obtain optimal order a priori error estimates for the approximate solution. All estimates are independent of the aspect ratio of the finite elements. Copyright © 2011 John Wiley & Sons, Ltd.