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

Langer, Ulrich; Zank, Marco

doi:10.1137/20m1358128

Cited by 10 publications

(20 citation statements)

References 30 publications

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“…i.e., it is essentially (up to the (log N ) 2 factor) equal to the number of degrees of freedom for the discretization of one spatial problem. Importantly, in the solution algorithms of [21], M O (log N ) 2 will reduce time and memory requirements.…”

Section: Conclusion Of the Proofmentioning

confidence: 99%

“…For the right-hand side G, the integrals for computing the projection Π M N g in (6.4) are calculated by using high-order quadrature rules. The global linear system (6.5) is solved in MATLAB by using the Bartels-Stewart method with real-Schur decomposition, see [21,Algorithm 4.1]. All calculations presented in this section were performed on a PC with two Intel Xeon E5-2687W v4 CPUs 3.00 GHz, i.e., in sum 24 cores and 512 GB main memory.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…It is of Petrov-Galerkin type, and is based on a fractional order Sobolev space in the temporal direction. It has been proposed and developed in a series of papers [34,35,39,21,33]. We briefly recapitulate it here, and refer to [34] for full development of details.…”

Section: Introductionmentioning

confidence: 99%

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

Perugia¹,

Schwab²,

Zank³

2022

Preprint

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For linear parabolic initial-boundary value problems with selfadjoint, 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 semidiscretization 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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Section: Conclusion Of the Proofmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

See 1 more Smart Citation

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

Perugia¹,

Schwab²,

Zank³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is of Petrov-Galerkin type, and is based on a fractional order Sobolev space in the temporal direction. It has been proposed and developed in a series of papers [22,[34][35][36]40]. We briefly recapitulate it here, and refer to [35] for full development of details.…”

Section: Introductionmentioning

confidence: 99%

Exponential convergence of hp-time-stepping in space-time discretizations of parabolic PDES

Perugia¹,

Schwab²,

Zank³

2023

ESAIM: M2AN

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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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“…Conforming space-time discretizations by piecewise polynomials of these variational formulations lead to huge linear systems. In [7,21,20], fast space-time solvers are developed, where some of them allow for an easy parallelization in time. Further extensions for this variational setting, which includes the modified Hilbert transformation H T , are an hp-FEM in temporal direction and a classical FEM with graded meshes in spatial direction, see [12].…”

Section: Introductionmentioning

confidence: 99%

Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation

Zank¹

2022

Preprint

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

Cited by 10 publications

References 30 publications

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

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

Exponential convergence of hp-time-stepping in space-time discretizations of parabolic PDES

Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation

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