For the discretisation of time-dependent partial differential equations, the classical approaches are time stepping schemes together with finite element methods in space. An alternative is to discretise the time-dependent problem without separating the temporal and spatial variables. However, space-time approximation methods depend strongly on the space-time variational formulations on the continuous level. The focus of this work is on space-time variational formulations for the heat and wave equation, which result not only in inf-sup stable formulations but fit also very well to conforming space-time discretisations.<\/p>\n
The first part investigates the heat equation in anisotropic Sobolev spaces, where a type of Hilbert transform is introduced such that ansatz and test spaces are equal. Unconditional stability is proven for any conforming discretisation of this space-time variational formulation.<\/p>\n
The second part considers space-time variational formulations for the wave equation. New existence and uniqueness results for the wave equation in a weak and in a strong sense are proven, including isomorphic solution operators and corresponding inf-sup conditions. In addition, an unconditionally stable space-time finite element method with piecewise linear, continuous functions is derived.<\/p>","protected":false},"excerpt":{"rendered":"
For the discretisation of time-dependent partial differential equations, the classical approaches are time stepping schemes together with finite element methods in space. An alternative is to discretise the time-dependent problem without separating the temporal and spatial variables. However, space-time approximation methods depend strongly on the space-time variational formulations on the continuous level. The focus of this work is on space-time variational formulations for the heat and wave equation, which result not only in inf-sup stable formulations but fit also very well to conforming space-time discretisations.<\/p>\n
The first part investigates the heat equation in anisotropic Sobolev spaces, where a type of Hilbert transform is introduced such that ansatz and test spaces are equal. Unconditional stability is proven for any conforming discretisation of this space-time variational formulation.<\/p>\n
The second part considers space-time variational formulations for the wave equation. New existence and uniqueness results for the wave equation in a weak and in a strong sense are proven, including isomorphic solution operators and corresponding inf-sup conditions. In addition, an unconditionally stable space-time finite element method with piecewise linear, continuous functions is derived.<\/p>\n","protected":false},"featured_media":40305,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"yoast_head":"\n\n\n\n\n\n\n\n\n\n\t\n\t\n\n\n\t\n