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12 papers
A note on the space-time variational formulation for the wave equation with source term in
Marco Zank
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 v…
A Space-Time Continuous Galerkin Finite Element Method for Linear Schrödinger Equations
Marco Zank
We introduce a space-time finite element method for the linear time-dependent Schrödinger equation with Dirichlet conditions in a bounded Lipschitz domain. The proposed discretizat…
On a modified Hilbert transformation, the discrete inf-sup condition, and error estimates
Richard Löscher, Olaf Steinbach, Marco Zank
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-dependen…
Wavelet compressed, modified Hilbert transform in the space-time discretization of the heat equation
Helmut Harbrecht, Christoph Schwab, Marco Zank
On a finite time interval , we consider the multiresolution Galerkin discretization of a modified Hilbert transform which arises in the space-time Galerkin di…
Numerical study of conforming space-time methods for Maxwell's equations
Julia I. M. Hauser, Marco Zank
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…
Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation
Marco Zank
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 elem…
Exponential Convergence of -Time-Stepping in Space-Time Discretizations of Parabolic PDEs
Ilaria Perugia, Christoph Schwab, Marco Zank
For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and fo…
Towards coercive boundary element methods for the wave equation
Olaf Steinbach, Carolina Urzúa-Torres, Marco Zank
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-know…
Numerical results for an unconditionally stable space-time finite element method for the wave equation
Richard Löscher, Olaf Steinbach, Marco Zank
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,…
Higher-Order Space-Time Continuous Galerkin Methods for the Wave Equation
Marco Zank
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-…
A generalized inf-sup stable variational formulation for the wave equation
Olaf Steinbach, Marco Zank
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…
Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces
Ulrich Langer, Marco Zank
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…