Marcus Grote - Academia.edu (original) (raw)

Papers by Marcus Grote

Journal of Computational and Applied Mathematics, Feb 1, 2013

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for ... more Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams-Bashforth multi-step methods, local time-stepping methods of arbitrarily high order of accuracy are derived for damped wave equations. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time-stepping methods.

arXiv (Cornell University), Jul 30, 2021

Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield sma... more Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield small subspaces where the characteristic functions comprising u are well approximated. When combined with Newton-like optimization methods for the solution of inverse medium problems, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space. Here, we derive L 2-error estimates for the AS decomposition of u, truncated after K terms, when u is piecewise constant and consists of K characteristic functions over Lipschitz domains and a background. Our estimates apply both to the continuous and the discrete Galerkin finite element setting. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory.

... It is shown to satisfy the smoothing property for symmetric positive definite problems. Numer... more ... It is shown to satisfy the smoothing property for symmetric positive definite problems. Numerical experiments show that SPAI-0 smoothing is usually preferable to damped Jacobi smooth-ing. ... and ح the upper triangular part of . Then damped Jacobi smoothing corresponds to ...

of the ordinary differential equation which occurs in the boundary condition. An exact nonreflect... more of the ordinary differential equation which occurs in the boundary condition. An exact nonreflecting boundary condition was derived previously for use with the time dependent wave equation in three Finally, we shall solve a sequence of scattering problems space dimensions. Here it is shown how to combine that boundary by using an explicit finite difference method and our condition with finite difference methods and finite element methboundary condition. We shall also solve the same problems ods. Uniqueness of the solution is proved, stability issues are disby using two of the standard artificial boundary conditions. cussed, and improvements are proposed for numerical computa-Comparison of these solutions with the ''exact'' solution, tion. Numerical examples are presented which demonstrate the improvement in accuracy over standard methods. ᮊ 1996 Academic obtained by computing in a very large domain so that Press, Inc. spurious reflections are postponed, shows that our boundary condition is much more accurate than the standard ones. Our boundary condition also has the advantage that

Springer eBooks, Jan 11, 2008

An exact nonre ecting boundary condition was derived previously for use with the time dependent M... more An exact nonre ecting boundary condition was derived previously for use with the time dependent Maxwell equations in three space dimensions 1. Here it is shown how to combine that boundary condition with the variational formulation for use with the nite element method. The fundamental theory underlying the derivation of the exact boundarycondition is reviewed. Numerical examples with the nite-di erence timedomain method are presented which demonstrate the high accuracy and con rm the expected rate of convergence of the numerical method.

Springer eBooks, 2003

A new far-field evaluation formula is pr esented which enables the efficient evaluation of the fa... more A new far-field evaluation formula is pr esented which enables the efficient evaluation of the far-field solution for wave propagation problems, if the exact nonreflecting boundary condition in [5, 6] is used for the numerical computation in the near-field. In particular, the evaluation formula permits to take advantage of the fast decay with distance of selected modes and thus to store only the minimal amount of information necessary from the past. The accuracy of this formula is illustrated via numerical experiments.

PPSC, 1995

Effective Parallel Preconditioning with Sparse Approximate Inverses* Marcus Grote* Thomas Huckle*... more Effective Parallel Preconditioning with Sparse Approximate Inverses* Marcus Grote* Thomas Huckle* Abstract A parallel preconditioner is presented for the ... We also wish to thank DavidSilvester and Howard Elman who provided us with the matrices coming from incompressible ...

IAHS-AISH publication, 2017

ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010, 2010

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for ... more Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local time-stepping methods are developed, which allow arbitrarily small time steps precisely where small elements in the mesh are located. When combined with a discontinuous Galerkin finite element discretization in space, which inherently leads to a diagonal mass matrix, the resulting numerical schemes are fully explicit. Starting from the classical Adams-Bashforth multi-step methods, local time stepping schemes of arbitrarily high accuracy are derived. Numerical experiments validate the theory and illustrate the usefulness of the proposed time integration schemes.

ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006, Sep 5, 2006

We propose efficient algebraic multilevel preconditioning for the Helmholtz equation with high wa... more We propose efficient algebraic multilevel preconditioning for the Helmholtz equation with high wave numbers. Our algebraic method is mainly based on using new multilevel incomplete LDL T techniques for symmetric indefinite systems.

De Gruyter eBooks, Sep 16, 2013

Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discon... more Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time integration is used, the time-step is constrained by the smallest elements in the mesh for numerical stability, possibly a high price to pay. To overcome that overly restrictive stability constraint on the time-step, yet without resorting to implicit methods, explicit local time-stepping schemes (LTS) are presented here for transient wave equations either with or without damping. In the undamped case, leapfrog based LTS methods lead to high-order explicit LTS schemes, which conserve the energy. In the damped case, when energy is no longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS schemes of arbitrarily high accuracy. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting timemarching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations validate the theory and illustrate the usefulness of these local time-stepping methods.

Comptes Rendus Mathematique, Jun 1, 2013

Numerical Analysis FE heterogeneous multiscale method for long-time wave propagation Méthode d'él... more Numerical Analysis FE heterogeneous multiscale method for long-time wave propagation Méthode d'éléments finis multi-échelles pour l'équation des ondes dans des milieux hétérogènes sur des temps longs

Computer Methods in Applied Mechanics and Engineering, Jun 1, 2006

An exact nonreflecting boundary condition is derived for the time dependent Maxwell equations in ... more An exact nonreflecting boundary condition is derived for the time dependent Maxwell equations in three space dimensions. It holds on a spherical surface B, outside of which the medium is assumed to be homogeneous, isotropic, and source-free. This boundary condition is local in space and time, and it does not involve high-order derivatives. Thus it is easy to incorporate into standard numerical methods. Numerical examples demonstrate the usefulness and high accuracy of this local nonreflecting boundary condition.

SIAM Journal on Scientific Computing, 2015

Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for th... more Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or low-storage explicit RK methods, explicit LTS methods of arbitrarily high accuracy are derived. When combined with an essentially diagonal finite element mass matrix, the resulting time-marching schemes retain the high accuracy, stability and efficiency of the original RK methods while circumventing the geometry-induced stiffness. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTS-RK methods.

Journal of Computational and Applied Mathematics, Oct 1, 2010

Explicit local time-stepping methods are derived for the time dependent Maxwell equations in cond... more Explicit local time-stepping methods are derived for the time dependent Maxwell equations in conducting and non-conducting media. By using smaller time steps precisely where smaller elements in the mesh are located, these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting discrete time-marching schemes are fullly explicit and thus inherently parallel. In a non-conducting source-free medium they also conserve a discrete energy, which provides a rigorous criterion for stability. Starting from the standard leapfrog scheme, local time-stepping methods of arbitrarily high accuracy are derived for non-conducting media. Numerical experiments with a discontinuous Galerkin discretization in space validate the theory and illustrate the usefulness of the proposed time integration schemes.

Journal of Scientific Computing, Sep 18, 2008

a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-depend... more a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L 2-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation ("leap-frog" scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1 + t 2), where p denotes the polynomial degree, h the mesh size, and t the time step.

Journal of Computational Physics, 1998

A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describ... more A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation.

Journal of Computational and Applied Mathematics, Feb 1, 2013

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for ... more Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams-Bashforth multi-step methods, local time-stepping methods of arbitrarily high order of accuracy are derived for damped wave equations. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these local time-stepping methods.

arXiv (Cornell University), Jul 30, 2021

Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield sma... more Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield small subspaces where the characteristic functions comprising u are well approximated. When combined with Newton-like optimization methods for the solution of inverse medium problems, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space. Here, we derive L 2-error estimates for the AS decomposition of u, truncated after K terms, when u is piecewise constant and consists of K characteristic functions over Lipschitz domains and a background. Our estimates apply both to the continuous and the discrete Galerkin finite element setting. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory.

... It is shown to satisfy the smoothing property for symmetric positive definite problems. Numer... more ... It is shown to satisfy the smoothing property for symmetric positive definite problems. Numerical experiments show that SPAI-0 smoothing is usually preferable to damped Jacobi smooth-ing. ... and ح the upper triangular part of . Then damped Jacobi smoothing corresponds to ...

of the ordinary differential equation which occurs in the boundary condition. An exact nonreflect... more of the ordinary differential equation which occurs in the boundary condition. An exact nonreflecting boundary condition was derived previously for use with the time dependent wave equation in three Finally, we shall solve a sequence of scattering problems space dimensions. Here it is shown how to combine that boundary by using an explicit finite difference method and our condition with finite difference methods and finite element methboundary condition. We shall also solve the same problems ods. Uniqueness of the solution is proved, stability issues are disby using two of the standard artificial boundary conditions. cussed, and improvements are proposed for numerical computa-Comparison of these solutions with the ''exact'' solution, tion. Numerical examples are presented which demonstrate the improvement in accuracy over standard methods. ᮊ 1996 Academic obtained by computing in a very large domain so that Press, Inc. spurious reflections are postponed, shows that our boundary condition is much more accurate than the standard ones. Our boundary condition also has the advantage that

Springer eBooks, Jan 11, 2008

An exact nonre ecting boundary condition was derived previously for use with the time dependent M... more An exact nonre ecting boundary condition was derived previously for use with the time dependent Maxwell equations in three space dimensions 1. Here it is shown how to combine that boundary condition with the variational formulation for use with the nite element method. The fundamental theory underlying the derivation of the exact boundarycondition is reviewed. Numerical examples with the nite-di erence timedomain method are presented which demonstrate the high accuracy and con rm the expected rate of convergence of the numerical method.

Springer eBooks, 2003

A new far-field evaluation formula is pr esented which enables the efficient evaluation of the fa... more A new far-field evaluation formula is pr esented which enables the efficient evaluation of the far-field solution for wave propagation problems, if the exact nonreflecting boundary condition in [5, 6] is used for the numerical computation in the near-field. In particular, the evaluation formula permits to take advantage of the fast decay with distance of selected modes and thus to store only the minimal amount of information necessary from the past. The accuracy of this formula is illustrated via numerical experiments.

PPSC, 1995

Effective Parallel Preconditioning with Sparse Approximate Inverses* Marcus Grote* Thomas Huckle*... more Effective Parallel Preconditioning with Sparse Approximate Inverses* Marcus Grote* Thomas Huckle* Abstract A parallel preconditioner is presented for the ... We also wish to thank DavidSilvester and Howard Elman who provided us with the matrices coming from incompressible ...

IAHS-AISH publication, 2017

ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010, 2010

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for ... more Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local time-stepping methods are developed, which allow arbitrarily small time steps precisely where small elements in the mesh are located. When combined with a discontinuous Galerkin finite element discretization in space, which inherently leads to a diagonal mass matrix, the resulting numerical schemes are fully explicit. Starting from the classical Adams-Bashforth multi-step methods, local time stepping schemes of arbitrarily high accuracy are derived. Numerical experiments validate the theory and illustrate the usefulness of the proposed time integration schemes.

ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006, Sep 5, 2006

We propose efficient algebraic multilevel preconditioning for the Helmholtz equation with high wa... more We propose efficient algebraic multilevel preconditioning for the Helmholtz equation with high wave numbers. Our algebraic method is mainly based on using new multilevel incomplete LDL T techniques for symmetric indefinite systems.

De Gruyter eBooks, Sep 16, 2013

Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discon... more Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time integration is used, the time-step is constrained by the smallest elements in the mesh for numerical stability, possibly a high price to pay. To overcome that overly restrictive stability constraint on the time-step, yet without resorting to implicit methods, explicit local time-stepping schemes (LTS) are presented here for transient wave equations either with or without damping. In the undamped case, leapfrog based LTS methods lead to high-order explicit LTS schemes, which conserve the energy. In the damped case, when energy is no longer conserved, Adams-Bashforth based LTS methods also lead to explicit LTS schemes of arbitrarily high accuracy. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting timemarching schemes are fully explicit and thus inherently parallel. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations validate the theory and illustrate the usefulness of these local time-stepping methods.

Comptes Rendus Mathematique, Jun 1, 2013

Numerical Analysis FE heterogeneous multiscale method for long-time wave propagation Méthode d'él... more Numerical Analysis FE heterogeneous multiscale method for long-time wave propagation Méthode d'éléments finis multi-échelles pour l'équation des ondes dans des milieux hétérogènes sur des temps longs

Computer Methods in Applied Mechanics and Engineering, Jun 1, 2006

An exact nonreflecting boundary condition is derived for the time dependent Maxwell equations in ... more An exact nonreflecting boundary condition is derived for the time dependent Maxwell equations in three space dimensions. It holds on a spherical surface B, outside of which the medium is assumed to be homogeneous, isotropic, and source-free. This boundary condition is local in space and time, and it does not involve high-order derivatives. Thus it is easy to incorporate into standard numerical methods. Numerical examples demonstrate the usefulness and high accuracy of this local nonreflecting boundary condition.

SIAM Journal on Scientific Computing, 2015

Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for th... more Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or low-storage explicit RK methods, explicit LTS methods of arbitrarily high accuracy are derived. When combined with an essentially diagonal finite element mass matrix, the resulting time-marching schemes retain the high accuracy, stability and efficiency of the original RK methods while circumventing the geometry-induced stiffness. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTS-RK methods.

Journal of Computational and Applied Mathematics, Oct 1, 2010

Explicit local time-stepping methods are derived for the time dependent Maxwell equations in cond... more Explicit local time-stepping methods are derived for the time dependent Maxwell equations in conducting and non-conducting media. By using smaller time steps precisely where smaller elements in the mesh are located, these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting discrete time-marching schemes are fullly explicit and thus inherently parallel. In a non-conducting source-free medium they also conserve a discrete energy, which provides a rigorous criterion for stability. Starting from the standard leapfrog scheme, local time-stepping methods of arbitrarily high accuracy are derived for non-conducting media. Numerical experiments with a discontinuous Galerkin discretization in space validate the theory and illustrate the usefulness of the proposed time integration schemes.

Journal of Scientific Computing, Sep 18, 2008

a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-depend... more a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L 2-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation ("leap-frog" scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1 + t 2), where p denotes the polynomial degree, h the mesh size, and t the time step.

Journal of Computational Physics, 1998

A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describ... more A new discrete non-reflecting boundary condition for the time-dependent Maxwell equations describing the propagation of an electromagnetic wave in an infinite homogenous lossless rectangular waveguide with perfectly conducting walls is presented. It is derived from a virtual spatial finite difference discretization of the problem on the unbounded domain. Fourier transforms are used to decouple transversal modes. A judicious combination of edge based nodal values permits us to recover a simple structure in the Laplace domain. Using this, it is possible to approximate the convolution in time by a similar fast convolution algorithm as for the standard wave equation.