arXiv subjects

Marco Zank

Publications and source records attributed to Marco Zank.

12 recordsLinked to original sources

arXiv2025

A note on the space-time variational formulation for the wave equation with source term in $L^2(Q)$

We derive a variational formulation for the scalar wave equation in the second-order formulation on bounded Lipschitz domains and homogeneous initial conditions. We investigate a variational framework in a bounded space-time cylinder $Q$ with a new solution space and the test space $L^2(Q)$ for source terms in $L^2(Q)$. Using existence and uniqueness results in $H^1(Q)$, we prove that this variational setting fits the inf-sup theory, including an isomorphism as solution operator. Moreover, we show that the new solution space is not a subspace of $H^2(Q)$. This new uniqueness and solvability result is not only crucial for discretizations using space-time methods, including least-squares approaches, but also important for regularity results and the analysis of related space-time boundary integral equations, which form the basis for space-time boundary element methods.

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arXiv2025

A Space-Time Continuous Galerkin Finite Element Method for Linear Schr\"odinger Equations

We introduce a space-time finite element method for the linear time-dependent Schr\"odinger equation with Dirichlet conditions in a bounded Lipschitz domain. The proposed discretization scheme is based on a space-time variational formulation of the time-dependent Schr\"odinger equation. In particular, the space-time method is conforming and is of Galerkin-type, i.e., trial and test spaces are equal. We consider a tensor-product approach with respect to time and space, using piecewise polynomial, continuous trial and test functions. In this case, we state the global linear system and efficient direct space-time solvers based on exploiting the Kronecker structure of the global system matrix. This leads to the Bartels-Stewart method and the fast diagonalization method. Both methods result in solving a sequence of spatial subproblems. In particular, the fast diagonalization method allows for solving the spatial subproblems in parallel, i.e., a time parallelization is possible. Numerical examples for a two-dimensional spatial domain illustrate convergence in space-time norms and show the potential of the proposed solvers.

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arXiv2024

Wavelet compressed, modified Hilbert transform in the space-time discretization of the heat equation

On a finite time interval $(0,T)$, we consider the multiresolution Galerkin discretization of a modified Hilbert transform $\mathcal H_T$ which arises in the space-time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in $(0,T)$ consisting of piecewise polynomials of degree $\geq 1$ with sufficiently many vanishing moments which constitute Riesz bases in the Sobolev spaces $H^{s}_{0,}(0,T)$ and $H^{s}_{,0}(0,T)$. These bases provide stable multilevel splittings of the temporal discretization spaces into "increment" or "detail" spaces. Furthermore, they allow to optimally compress the nonlocal integrodifferential operators which appear in stable space-time variational formulations of initial-boundary value problems, such as the heat equation and the acoustic wave equation. We then obtain sparse space-time tensor-product spaces via algebraic tensor-products of the temporal multilevel discretizations with standard, hierarchic finite element spaces in the spatial domain (with standard Lagrangian FE bases). Hence, the construction of multiresolutions in the spatial domain is not necessary. An efficient multilevel preconditioner is proposed that solves the linear system of equations resulting from the sparse space-time Galerkin discretization with essentially linear complexity (in work and memory). A substantial reduction in the number of the degrees of freedom and CPU time (compared to time-marching discretizations) is demonstrated in numerical experiments.

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arXiv2024

On a modified Hilbert transformation, the discrete inf-sup condition, and error estimates

In this paper, we analyze the discrete inf-sup condition and related error estimates for a modified Hilbert transformation as used in the space-time discretization of time-dependent partial differential equations. It turns out that the stability constant depends linearly on the finite element mesh parameter, but in most cases, we can show optimal convergence. We present a series of numerical experiments which illustrate the theoretical findings.

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arXiv2023

Numerical study of conforming space-time methods for Maxwell's equations

Time-dependent Maxwell's equations govern electromagnetics. Under certain conditions, we can rewrite these equations into a partial differential equation of second order, which in this case is the vectorial wave equation. For the vectorial wave, we investigate the numerical application and the challenges in the implementation. For this purpose, we consider a space-time variational setting, i.e. time is just another spatial dimension. More specifically, we apply integration by parts in time as well as in space, leading to a space-time variational formulation with different trial and test spaces. Conforming discretizations of tensor-product type result in a Galerkin--Petrov finite element method that requires a CFL condition for stability. For this Galerkin--Petrov variational formulation, we study the CFL condition and its sharpness. To overcome the CFL condition, we use a Hilbert-type transformation that leads to a variational formulation with equal trial and test spaces. Conforming space-time discretizations result in a new Galerkin--Bubnov finite element method that is unconditionally stable. In numerical examples, we demonstrate the effectiveness of this Galerkin--Bubnov finite element method. Furthermore, we investigate different projections of the right-hand side and their influence on the convergence rates. This paper is the first step towards a more stable computation and a better understanding of vectorial wave equations in a conforming space-time approach.

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arXiv2022

Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation

We present quadrature schemes to calculate matrices, where the so-called modified Hilbert transformation is involved. These matrices occur as temporal parts of Galerkin finite element discretizations of parabolic or hyperbolic problems when the modified Hilbert transformation is used for the variational setting. This work provides the calculation of these matrices to machine precision for arbitrary polynomial degrees and non-uniform meshes. The proposed quadrature schemes are based on weakly singular integral representations of the modified Hilbert transformation. First, these weakly singular integral representations of the modified Hilbert transformation are proven. Second, using these integral representations, we derive quadrature schemes, which treat the occurring singularities appropriately. Thus, exponential convergence with respect to the number of quadrature nodes for the proposed quadrature schemes is achieved. Numerical results, where this exponential convergence is observed, conclude this work.

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arXiv2022

Exponential Convergence of $hp$-Time-Stepping in Space-Time Discretizations of Parabolic PDEs

For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains $D$, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete $hp$-time-stepping approach. We combine this semi-discretization in time with first-order, so-called "$h$-version" Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of $D$, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in $D$. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.

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arXiv2021

Towards coercive boundary element methods for the wave equation

In this note, we discuss the ellipticity of the single layer boundary integral operator for the wave equation in one space dimension. This result not only generalizes the well-known ellipticity of the energetic boundary integral formulation in $L^2$, but it also turns out to be a particular case of a recent result on the inf-sup stability of boundary integral operators for the wave equation. Instead of the time derivative in the energetic formulation, we use a modified Hilbert transformation, which allows us to stay in Sobolev spaces of the same order. This results in the applicability of standard boundary element error estimates, which are confirmed by numerical results.

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arXiv2021

Numerical results for an unconditionally stable space-time finite element method for the wave equation

In this work, we introduce a new space-time variational formulation of the second-order wave equation, where integration by parts is also applied with respect to the time variable, and a modified Hilbert transformation is used. For this resulting variational setting, ansatz and test spaces are equal. Thus, conforming finite element discretizations lead to Galerkin--Bubnov schemes. We consider a conforming tensor-product approach with piecewise polynomial, continuous basis functions, which results in an unconditionally stable method, i.e., no CFL condition is required. We give numerical examples for a one- and a two-dimensional spatial domain, where the unconditional stability and optimal convergence rates in space-time norms are illustrated.

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arXiv2021

Higher-Order Space-Time Continuous Galerkin Methods for the Wave Equation

We consider a space-time variational formulation of the second-order wave equation, where integration by parts is also applied with respect to the time variable. Conforming tensor-product finite element discretisations with piecewise polynomials of this space-time variational formulation require a CFL condition to ensure stability. To overcome this restriction in the case of piecewise multilinear, continuous ansatz and test functions, a stabilisation is well-known, which leads to an unconditionally stable space-time finite element method. In this work, we generalise this stabilisation idea from the lowest-order case to the higher-order case, i.e. to an arbitrary polynomial degree. We give numerical examples for a one-dimensional spatial domain, where the unconditional stability and optimal convergence rates in space-time norms are illustrated.

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arXiv2021

A generalized inf-sup stable variational formulation for the wave equation

In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of $H^1(Q$) with $Q$ being the space-time domain, the classical assumption is to consider the right-hand side $f$ in $L^2(Q)$. Here, we analyze a generalized setting of this variational formulation, which allows us to prove unique solvability also for $f$ being in the dual space of the test space, i.e., the solution operator is an isomorphism between the ansatz space and the dual of the test space. This new approach is based on a suitable extension of the ansatz space to include the information of the differential operator of the wave equation at the initial time $t=0$. These results are of utmost importance for the formulation and numerical analysis of unconditionally stable space-time finite element methods, and for the numerical analysis of boundary element methods to overcome the well-known norm gap in the analysis of boundary integral operators.

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arXiv2020

Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces

We consider a space-time variational formulation of parabolic initial-boundary value problems in anisotropic Sobolev spaces in combination with a Hilbert-type transformation. This variational setting is the starting point for the space-time Galerkin finite element discretization that leads to a large global linear system of algebraic equations. We propose and investigate new efficient direct solvers for this system. In particular, we use a tensor-product approach with piecewise polynomial, globally continuous ansatz and test functions. The developed solvers are based on the Bartels-Stewart method and on the Fast Diagonalization method, which result in solving a sequence of spatial subproblems. The solver based on the Fast Diagonalization method allows to solve these spatial subproblems in parallel leading to a full parallelization in time. We analyze the complexity of the proposed algorithms, and give numerical examples for a two-dimensional spatial domain, where sparse direct solvers for the spatial subproblems are used.

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