A matrix-based approach to the stability of a space-time isogeometric method for the linear Schr\"odinger equation

Matteo Ferrari; Sergio G\'omez

arxiv: 2506.18859 · v2 · submitted 2025-06-23 · 🧮 math.NA · cs.NA

A matrix-based approach to the stability of a space-time isogeometric method for the linear Schr\"odinger equation

Matteo Ferrari , Sergio G\'omez This is my paper

Pith reviewed 2026-05-19 07:46 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords isogeometric analysisspace-time methodsSchrödinger equationstability analysisToeplitz matricesunconditional stabilitymass conservationenergy conservation

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The pith

A matrix analysis of nearly Toeplitz systems establishes unconditional stability for a space-time isogeometric discretization of the linear Schrödinger equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a space-time isogeometric finite element method for the linear Schrödinger equation that employs maximal-regularity splines in time. Because these splines have nonlocal support, conventional variational coercivity arguments cannot be applied to prove stability. The authors instead examine the discrete linear systems directly and demonstrate that the governing matrices form a family of nearly Toeplitz operators. By bounding the condition numbers of these matrices through symbol analysis, they establish that the family is weakly well-conditioned, which implies unconditional stability for any time-step size. The same analysis shows that the discrete solution exactly conserves mass and energy at the final time.

Core claim

The family of nearly Toeplitz system matrices that arise from the space-time isogeometric discretization remains weakly well-conditioned independently of the time-step size. This property guarantees unconditional stability of the scheme. The discrete solution preserves mass and energy at the final time. The same matrix-based technique supplies a complete stability proof for a related first-order-in-time space-time isogeometric method for the wave equation.

What carries the argument

Nearly Toeplitz system matrices whose condition numbers are controlled by symbol analysis

If this is right

The scheme remains stable for arbitrary time-step sizes.
Mass and energy of the solution are exactly preserved at the final time.
Optimal convergence rates are attained in both space and time.
A complete matrix-based stability analysis is obtained for the analogous space-time isogeometric method applied to the wave equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The matrix-symbol technique may extend to other space-time discretizations that use nonlocal bases for dispersive problems.
The near-Toeplitz structure could be exploited to construct fast iterative or direct solvers for large three-dimensional simulations.
Similar condition-number arguments might apply to nonlinear Schrödinger equations or to systems with variable coefficients.

Load-bearing premise

The discrete system matrices remain nearly Toeplitz with the same symbol structure when maximal-regularity splines of arbitrary degree are used in time.

What would settle it

A direct numerical computation of the eigenvalues of the system matrix for successively higher spline degrees or smaller time steps that reveals condition numbers growing without bound.

Figures

Figures reproduced from arXiv: 2506.18859 by Matteo Ferrari, Sergio G\'omez.

**Figure 2.** Figure 2: Relative errors in the norms defined in (5.3), with maximal regularity splines in both space and time for polynomial degrees p = 3 ( marker), p = 4 (■ marker) and p = 5 (♦ marker), and the exact solution as in (5.2) with ω = 10 and n = 2. Here, ht = 0.0625 and hx decreases. Convergence. In [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: Relative errors in the H1 (QT ) norm (left plot) and the L 2 (QT ) norm (right plot) for varying ht = hx with maximal regularity splines in space and time with polynomial degree p = 1 (• marker), p = 2 (▲ marker), p = 3 ( marker), p = 4 (■ marker) and p = 5 (♦ marker). The exact solution is as in (5.2) with ω = 10 and n = 2. Conservation. Due to the rapid decay of the exact solution close to the boundary ∂… view at source ↗

**Figure 4.** Figure 4: Relative errors of the energy (left plot) and mass (right plot) functionals with maximal regularity [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

read the original abstract

We propose a space-time isogeometric finite element method for the linear Schr\"odinger equation, and establish its unconditional stability through a matrix-based analysis. Although maximal-regularity splines in time provide higher accuracy per degree of freedom compared to piecewise continuous polynomials, the nonlocal support of the spline bases precludes the use of standard variational arguments in the stability proofs. To overcome this, we show that the resulting scheme is governed by a family of nearly Toeplitz system matrices and, by studying the condition number of these matrices, we prove that the family is weakly well-conditioned, which guarantees the unconditional stability of the method. Furthermore, the discrete scheme preserves mass and energy at the final time. Numerical experiments confirm our theoretical findings and illustrate the optimal convergence behavior of the scheme. Finally, we exploit an algebraic connection between our formulation and a recent first-order-in-time space-time isogeometric method for the wave equation to derive a complete matrix-based stability analysis for the latter.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper establishes unconditional stability for a space-time IGA Schrödinger solver through nearly Toeplitz matrix conditioning, with an extension to the wave equation.

read the letter

Dear colleague, The main point of this paper is a matrix-based proof that a space-time isogeometric method for the linear Schrödinger equation is unconditionally stable. They use maximal-regularity B-splines in time, which are nonlocal, so standard variational stability arguments do not apply directly. Instead they show that the discrete operators lead to nearly Toeplitz matrices and that these matrices are weakly well-conditioned, meaning their condition numbers are bounded independently of the mesh size. This gives the stability. They also show mass and energy are preserved at the final time, and they use an algebraic connection to obtain a similar stability result for a space-time method for the wave equation. The paper does a good job making the algebraic reduction explicit and backing it with numerical experiments that show the expected convergence rates. The connection to the wave equation is a useful byproduct that extends the reach of the analysis without much additional effort. One area to watch is the dependence on the spline degree. The nearly Toeplitz structure comes from the local support of the basis functions, but with maximal regularity the continuity is high and the support width is p+1. The stress-test note points out that if the deviation from Toeplitz grows with p, the conditioning bound might not stay uniform. The paper claims the family is weakly well-conditioned, so presumably they control the perturbation terms, but it would be worth verifying that the symbol analysis and the bound hold for arbitrary p rather than just small fixed degrees. This work is aimed at people in the isogeometric analysis community who deal with time-dependent problems or space-time formulations. Anyone looking for stability proofs that rely on matrix properties instead of variational coercivity will find it relevant. The thinking is clear and the results are checkable, so it deserves a serious referee. I would send it for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a space-time isogeometric finite element method for the linear Schrödinger equation using maximal-regularity B-splines in time. It establishes unconditional stability through a matrix-based analysis by showing that the discrete system matrices belong to a family of nearly Toeplitz matrices that are weakly well-conditioned (with a bound independent of mesh size). This circumvents standard variational arguments precluded by the nonlocal support of the splines. The scheme preserves mass and energy at the final time. An algebraic connection is used to obtain a complete matrix-based stability analysis for a related first-order-in-time space-time IGA method for the wave equation. Numerical experiments confirm the theoretical results and demonstrate optimal convergence.

Significance. If the weak well-conditioning bound is shown to be uniform in both mesh size h and spline degree p, the matrix-based approach provides a useful alternative stability proof for space-time IGA schemes where variational coercivity arguments fail due to nonlocality. The mass/energy preservation is a clean algebraic feature, and the extension to the wave equation broadens the impact. The numerical confirmation of optimal rates is a strength.

major comments (2)

[Matrix analysis and symbol analysis sections] The unconditional stability claim rests on the nearly Toeplitz family being weakly well-conditioned with a condition-number bound independent of both h and p. The analysis must explicitly address whether the deviation from exact Toeplitz form (arising from the p+1 support width and C^{p-1} continuity of maximal-regularity splines) remains O(1) uniformly in p, or whether boundary/overlap perturbations grow with p. Please state the precise definition of 'weakly well-conditioned' and the symbol-analysis result (including any theorem establishing the bound) that guarantees uniformity in p.
[Preservation properties section] The mass and energy preservation at final time is presented as a separate algebraic identity. Confirm that this identity holds without relying on the conditioning bound or stability result, and clarify whether it is proven for arbitrary p.

minor comments (2)

[Abstract] The abstract introduces 'weakly well-conditioned' without a brief inline definition or reference to the precise bound; adding one sentence would improve accessibility.
[Numerical experiments] Numerical experiments should include runs with increasing spline degree p (e.g., p=2 to p=8) to empirically support uniformity of the stability bound with respect to p.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of the matrix-based stability analysis. We address each major comment below.

read point-by-point responses

Referee: [Matrix analysis and symbol analysis sections] The unconditional stability claim rests on the nearly Toeplitz family being weakly well-conditioned with a condition-number bound independent of both h and p. The analysis must explicitly address whether the deviation from exact Toeplitz form (arising from the p+1 support width and C^{p-1} continuity of maximal-regularity splines) remains O(1) uniformly in p, or whether boundary/overlap perturbations grow with p. Please state the precise definition of 'weakly well-conditioned' and the symbol-analysis result (including any theorem establishing the bound) that guarantees uniformity in p.

Authors: We thank the referee for highlighting the need for explicit clarification on p-uniformity. In the manuscript, a matrix family is called weakly well-conditioned when its condition number admits a bound independent of the mesh size h. The symbol analysis (Section 4) derives this bound from the limiting symbol of the interior Toeplitz blocks, with the deviation from exact Toeplitz structure controlled by the fixed support width p+1 and the C^{p-1} continuity; these perturbations remain O(1) with respect to p because the overlap and boundary contributions are localized and do not accumulate with increasing degree. The resulting eigenvalue bounds are therefore independent of h for each fixed p. We acknowledge that an explicit statement confirming the p-independence of the overall constant was not included and will add a short remark together with a reference to the symbol-analysis theorem in the revised version. revision: yes
Referee: [Preservation properties section] The mass and energy preservation at final time is presented as a separate algebraic identity. Confirm that this identity holds without relying on the conditioning bound or stability result, and clarify whether it is proven for arbitrary p.

Authors: The mass and energy preservation identities are derived directly from the algebraic structure of the discrete system (specifically, the skew-Hermitian character of the time-coupling matrix combined with the L2-orthogonality properties of the B-spline basis at the final time). The proof is purely algebraic and makes no reference to the conditioning bound or the stability result. The argument holds for arbitrary spline degree p, as it relies only on the partition-of-unity property and the integration-by-parts identities satisfied by maximal-regularity splines, both of which are valid independently of p. We will insert an explicit sentence in the revised manuscript confirming this independence. revision: partial

Circularity Check

0 steps flagged

No circularity: stability follows from explicit matrix structure and symbol analysis

full rationale

The derivation establishes the nearly Toeplitz character of the space-time system matrices directly from the isogeometric assembly with maximal-regularity B-splines, then obtains the weak well-conditioning bound by symbol analysis of those matrices; unconditional stability is a consequence of the bound rather than an input to the matrix classification. Mass and energy preservation at final time is shown via a separate algebraic identity on the discrete operators. No step reduces a claimed prediction to a fitted quantity or to a self-citation whose validity is presupposed; the argument is self-contained against the explicit stencil structure and does not invoke uniqueness theorems or ansatzes imported from prior work by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of B-splines and Toeplitz matrices; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption B-spline bases of maximal regularity generate system matrices whose symbols admit a uniform bound on the condition number independent of mesh size and time step.
Invoked to replace variational coercivity arguments when support is nonlocal.

pith-pipeline@v0.9.0 · 5703 in / 1425 out tokens · 23493 ms · 2026-05-19T07:46:01.230290+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We show that the resulting scheme is governed by a family of nearly Toeplitz system matrices and, by studying the condition number of these matrices, we prove that the family is weakly well-conditioned
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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the symbol polynomial qKp(ρ) ... has exactly two zeros of unitary modulus

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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Amodio and L

P. Amodio and L. Brugnano. The conditioning of Toeplitz band matrices.Math. Comput. Modelling, 23(10):29–42, 1996. 16

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A. K. Aziz and P. Monk. Continuous finite elements in space and time for the heat equation.Math. Comp., 52(186):255–274, 1989

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R. H. Bartels and G. W. Stewart. Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4]. Commun. ACM, 15(9):820–826, September 1972

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L. Demkowicz, J. Gopalakrishnan, S. Nagaraj, and P. Sep´ ulveda. A spacetime DPG method for the Schr¨ odinger equation.SIAM J. Numer. Anal., 55(4):1740–1759, 2017

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C. D¨ oding and P. Henning. UniformL∞-bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation.IMA J. Numer. Anal., 44(5):2892–2935, 2024

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S.-E. Ekstr¨ om, I. Furci, C. Garoni, C. Manni, S. Serra-Capizzano, and H. Speleers. Are the eigenvalues of the B-spline isogeometric analysis approximation of −∆u = λu known in almost closed form?Numer. Linear Algebra Appl., 25(5):e2198, 34, 2018

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M. Ferrari and S. Fraschini. Stability of conforming space–time isogeometric methods for the wave equation. Math. Comp., 2025. doi: 10.1090/mcom/4062

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Ferrari, S

M. Ferrari, S. Fraschini, G. Loli, and I. Perugia. Unconditionally stable space-time isogeometric discretization for the wave equation in Hamiltonian formulation. Accepted for publication in ESAIM Math. Model. Numer. Anal., 2025

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Ferrari and I

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Fraschini, G

S. Fraschini, G. Loli, A. Moiola, and G. Sangalli. An unconditionally stable space-time isogeometric method for the acoustic wave equation.Comput. Math. Appl., 169:205–222, 2024

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Garoni, C

C. Garoni, C. Manni, F. Pelosi, S. Serra-Capizzano, and H. Speleers. On the spectrum of stiffness matrices arising from isogeometric analysis.Numer. Math., 127(4):751–799, 2014

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G´ omez and A

S. G´ omez and A. Moiola. A space-time Trefftz discontinuous Galerkin method for the linear Schr¨ odinger equation.SIAM J. Numer. Anal., 60(2):688–714, 2022

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G´ omez and A

S. G´ omez and A. Moiola. A space-time DG method for the Schr¨ odinger equation with variable potential. Adv. Comput. Math., 50(2):Paper No. 15, 34, 2024

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G´ omez, A

S. G´ omez, A. Moiola, I. Perugia, and P. Stocker. On polynomial Trefftz spaces for the linear time-dependent Schr¨ odinger equation.Appl. Math. Lett., 146:Paper No. 108824, 10, 2023

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S. Hain and K. Urban. An ultra-weak space-time variational formulation for the Schr¨ odinger equation.J. Complexity, 85:Paper No. 101868, 22, 2024

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Karakashian and C

O. Karakashian and C. Makridakis. A space-time finite element method for the nonlinear Schr¨ odinger equation: the discontinuous Galerkin method.Math. Comp., 67(222):479–499, 1998

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Karakashian and C

O. Karakashian and C. Makridakis. A space-time finite element method for the nonlinear Schr¨ odinger equation: the continuous Galerkin method.SIAM J. Numer. Anal., 36(6):1779–1807, 1999. 17

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Langer and M

U. Langer and M. Zank. Efficient direct space-time finite element solvers for parabolic initial-boundary value problems in anisotropic Sobolev spaces.SIAM J. Sci. Comput., 43(4):A2714–A2736, 2021

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J.-L. Lions and E. Magenes.Non-homogeneous boundary value problems and applications. Vol. I, volume Band 181 ofDie Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York-Heidelberg,

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Lions and E

J.-L. Lions and E. Magenes.Non-homogeneous boundary value problems and applications. Vol. II, volume Band 182 ofDie Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York-Heidelberg,

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Translated from the French by P. Kenneth

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F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors.NIST handbook of mathematical functions. Cambridge: Cambridge University Press, 2010

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M. Tani. A preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysis.Comput. Math. Appl., 74(7):1690–1702, 2017

work page 2017
[31]

M. Zank. A space-time continuous galerkin finite element method for linear schr¨ odinger equations, 2025. Accepted for publication in the proceedings of the LSSC’25. A Auxiliary result Lemma A.1.Consider an invertible2×2block matrix M= B−C C B withB , C ∈R n×n. Suppose also thatBis invertible, and define the Schur complementS=B+C B −1C. Then, it holds κ(S...

work page 2025

[1] [1]

Amodio and L

P. Amodio and L. Brugnano. The conditioning of Toeplitz band matrices.Math. Comput. Modelling, 23(10):29–42, 1996. 16

work page 1996

[2] [2]

A. K. Aziz and P. Monk. Continuous finite elements in space and time for the heat equation.Math. Comp., 52(186):255–274, 1989

work page 1989

[3] [3]

R. H. Bartels and G. W. Stewart. Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4]. Commun. ACM, 15(9):820–826, September 1972

work page 1972

[4] [4]

Bressan, A

A. Bressan, A. Kushova, G. Sangalli, and M. Tani. Space-time least squares approximation for Schr¨ odinger equation and efficient solver. arXiv:2311.18461, 2023

work page arXiv 2023

[5] [5]

Brezis.Functional analysis, Sobolev spaces and partial differential equations

H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universitext. New York, NY: Springer, 2011

work page 2011

[6] [6]

C. de Boor. On calculating withB-splines.J. Approximation Theory, 6:50–62, 1972

work page 1972

[7] [7]

Demkowicz, J

L. Demkowicz, J. Gopalakrishnan, S. Nagaraj, and P. Sep´ ulveda. A spacetime DPG method for the Schr¨ odinger equation.SIAM J. Numer. Anal., 55(4):1740–1759, 2017

work page 2017

[8] [8]

D¨ oding and P

C. D¨ oding and P. Henning. UniformL∞-bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation.IMA J. Numer. Anal., 44(5):2892–2935, 2024

work page 2024

[9] [9]

Ekstr¨ om, I

S.-E. Ekstr¨ om, I. Furci, C. Garoni, C. Manni, S. Serra-Capizzano, and H. Speleers. Are the eigenvalues of the B-spline isogeometric analysis approximation of −∆u = λu known in almost closed form?Numer. Linear Algebra Appl., 25(5):e2198, 34, 2018

work page 2018

[10] [10]

M. Ferrari. XT-Schr¨ odinger.https://github.com/MatteoFerrari11/XT-Schrodinger.git, 2025

work page 2025

[11] [11]

Ferrari and S

M. Ferrari and S. Fraschini. Stability of conforming space–time isogeometric methods for the wave equation. Math. Comp., 2025. doi: 10.1090/mcom/4062

work page doi:10.1090/mcom/4062 2025

[12] [12]

Ferrari, S

M. Ferrari, S. Fraschini, G. Loli, and I. Perugia. Unconditionally stable space-time isogeometric discretization for the wave equation in Hamiltonian formulation. Accepted for publication in ESAIM Math. Model. Numer. Anal., 2025

work page 2025

[13] [13]

Ferrari and I

M. Ferrari and I. Perugia. Intrinsic unconditional stability in space-time isogeometric approximation of the acoustic wave equation in second-order formulation. arXiv:2503.11166, 2025

work page arXiv 2025

[14] [14]

Fraschini, G

S. Fraschini, G. Loli, A. Moiola, and G. Sangalli. An unconditionally stable space-time isogeometric method for the acoustic wave equation.Comput. Math. Appl., 169:205–222, 2024

work page 2024

[15] [15]

Garoni, C

C. Garoni, C. Manni, F. Pelosi, S. Serra-Capizzano, and H. Speleers. On the spectrum of stiffness matrices arising from isogeometric analysis.Numer. Math., 127(4):751–799, 2014

work page 2014

[16] [16]

G´ omez and A

S. G´ omez and A. Moiola. A space-time Trefftz discontinuous Galerkin method for the linear Schr¨ odinger equation.SIAM J. Numer. Anal., 60(2):688–714, 2022

work page 2022

[17] [17]

G´ omez and A

S. G´ omez and A. Moiola. A space-time DG method for the Schr¨ odinger equation with variable potential. Adv. Comput. Math., 50(2):Paper No. 15, 34, 2024

work page 2024

[18] [18]

G´ omez, A

S. G´ omez, A. Moiola, I. Perugia, and P. Stocker. On polynomial Trefftz spaces for the linear time-dependent Schr¨ odinger equation.Appl. Math. Lett., 146:Paper No. 108824, 10, 2023

work page 2023

[19] [19]

D. J. Griffiths.Introduction to quantum mechanics. Englewood Cliffs, NJ: Prentice Hall, 1995

work page 1995

[20] [20]

Hain and K

S. Hain and K. Urban. An ultra-weak space-time variational formulation for the Schr¨ odinger equation.J. Complexity, 85:Paper No. 101868, 22, 2024

work page 2024

[21] [21]

R. A. Horn and C. R. Johnson.Matrix analysis. Cambridge University Press, Cambridge, second edition, 2013

work page 2013

[22] [22]

Karakashian and C

O. Karakashian and C. Makridakis. A space-time finite element method for the nonlinear Schr¨ odinger equation: the discontinuous Galerkin method.Math. Comp., 67(222):479–499, 1998

work page 1998

[23] [23]

Karakashian and C

O. Karakashian and C. Makridakis. A space-time finite element method for the nonlinear Schr¨ odinger equation: the continuous Galerkin method.SIAM J. Numer. Anal., 36(6):1779–1807, 1999. 17

work page 1999

[24] [24]

Langer and M

U. Langer and M. Zank. Efficient direct space-time finite element solvers for parabolic initial-boundary value problems in anisotropic Sobolev spaces.SIAM J. Sci. Comput., 43(4):A2714–A2736, 2021

work page 2021

[25] [25]

Lions and E

J.-L. Lions and E. Magenes.Non-homogeneous boundary value problems and applications. Vol. I, volume Band 181 ofDie Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York-Heidelberg,

work page

[26] [27]

Lions and E

J.-L. Lions and E. Magenes.Non-homogeneous boundary value problems and applications. Vol. II, volume Band 182 ofDie Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York-Heidelberg,

work page

[27] [28]

Translated from the French by P. Kenneth

work page

[28] [29]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors.NIST handbook of mathematical functions. Cambridge: Cambridge University Press, 2010

work page 2010

[29] [30]

M. Tani. A preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysis.Comput. Math. Appl., 74(7):1690–1702, 2017

work page 2017

[30] [31]

M. Zank. A space-time continuous galerkin finite element method for linear schr¨ odinger equations, 2025. Accepted for publication in the proceedings of the LSSC’25. A Auxiliary result Lemma A.1.Consider an invertible2×2block matrix M= B−C C B withB , C ∈R n×n. Suppose also thatBis invertible, and define the Schur complementS=B+C B −1C. Then, it holds κ(S...

work page 2025