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Stability of Galerkin discretizations of a mixed space-time variational formulation of parabolic evolution equations
Rob Stevenson, Jan Westerdiep
TL;DR
Noncoercive space-time formulations make uniformly stable Galerkin trial-test pairs difficult to construct, limiting the route to quasi-optimal approximations. The paper introduces a simpler mixed formulation and analyzes its discretizations. For suitable spaces, the new method yields quasi-optimal approximations at lower computational cost than Andreev’s method, while Steinbach’s method can be far from quasi-optimal in the natural norm.
Problem
Known well-posed simultaneous space-time formulations using only local differential operators are generally noncoercive, making uniformly stable discrete trial-test pairs difficult to find.
Source: §1 · p. 1Method
Results
For suitable trial spaces, the new method yields quasi-optimal approximations at lower computational cost than Andreev’s method, while Steinbach’s method can be far from quasi-optimal in the natural mesh-independent norm.
Source: §6 · pp. 18–19Takeaways & Limitations
The new mixed formulation provides quasi-optimal Galerkin approximations for suitable trial spaces at lower cost than Andreev’s method.
Source: §6 · p. 19For Andreev’s formulation, the error bound requires stronger solution regularity than optimal best-approximation rates in its mesh-dependent norm and does not establish quasi-best approximation.
Source: §3.3 · p. 10
Abstract
from arXiv · showhide
We analyze Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space-time discretization methods introduced in [IMA J. Numer. Anal., 33(1) (2013), pp. 242-260] by R. Andreev and in [Comput. Methods Appl. Math., 15(4) (2015), pp. 551-566] by O. Steinbach.
1. INTRODUCTION
Simultaneous space-time methods offer parallelism and joint space-time adaptivity, but noncoercive formulations make uniformly stable trial-test pairs difficult to construct. This work introduces a simpler mixed system with stable Galerkin discretizations for suitable spaces.
1. INTRODUCTION
- Space-time methods are better suited than classical time marching for massive parallel implementation and may drive adaptivity simultaneously in space and time.Known well-posed local-operator formulations are generally noncoercive, making uniformly stable discrete trial-test pairs difficult to find; such stability is sufficient and necessary for quasi-optimality.Source: §1 · p. 1
- Steinbach’s method establishes stability in a weaker mesh-dependent norm for equal trial and test spaces, but some solutions are far from quasi-optimal in the natural mesh-independent norm.Andreev’s minimal-residual method was developed in response to the difficulty of constructing stable trial-test pairs.Source: §1 · pp. 1–2
- The new mixed system modifies Andreev’s approach, uses a smaller trial space for the secondary variable, and yields stable discretizations with sparser stiffness matrices and comparable numerical errors.It is constructed from a space-time variational formulation following the Brézis-Ekeland-Nayroles principle.Source: §1 · p. 2
1.1. Organization.
- The paper derives two mixed formulations, gives stability conditions and error bounds, establishes a stability condition for prismatic elements under mesh and projection assumptions, and presents one-dimensional numerical experiments.The notation C ≲ D means C is bounded by a multiple of D independently of the parameters on which C and D may depend.Source: §1.1 · p. 2, §1.2 · p. 2
2. SPACE-TIME FORMULATIONS OF THE PARABOLIC EVOLUTION PROBLEM
The parabolic problem is recast in well-posed space-time variational and self-adjoint mixed forms. Their Galerkin discretizations differ in stability requirements and in the approximation demands placed on secondary variables.
2. SPACE-TIME FORMULATIONS OF THE PARABOLIC EVOLUTION PROBLEM
- The parabolic setting assumes a Gelfand triple V ↪ H ≃ H′ ↪ V′ and measurable time-dependent bilinear forms; a Gårding inequality can be shifted to coercivity by transforming the problem.The spaces X and Y are equipped with energy norms equivalent to their standard norms.Source: §2 · pp. 3, 5
- The variational problem is well-posed when the initial condition is included through the trace term alongside the space-time operator equation.The trace map is continuous from X into H, supporting the initial-value term.Source: §2 · p. 3
- Applying the Brézis-Ekeland-Nayroles principle gives an equivalent well-posed formulation whose left-hand operator is boundedly invertible, self-adjoint, and coercive.The parabolic problem seeks u satisfying ∂t u(t) + A(t)u(t) = g(t) with initial value u(0) = u0.Source: §2 · pp. 3–4
- The paper presents a self-adjoint saddle-point formulation with auxiliary variables and a second mixed formulation whose operator is boundedly invertible.The latter is obtained as a Schur complement; the secondary variable is nonzero, unlike the auxiliary variables in the first system.Source: §2 · pp. 4–5
- The mixed systems are equivalent at the continuous level, but their Galerkin discretizations are no longer equivalent.The new system’s secondary variable λ must be approximated, and its best-approximation error affects the primal-variable error bound.Source: §2 · p. 5
- For the new mixed system, stability is easier to realize, holds also when Aa ≠ 0, and requires fewer unknowns; its system matrix is sparser.The secondary variable λ is nonzero, so its approximation space must have approximation properties, and its best-approximation error enters the upper bound for the primal variable.Source: §2 · p. 5
3. STABLE DISCRETIZATIONS OF THE PARABOLIC PROBLEM
This section analyzes Galerkin discretizations of the parabolic problem, relating uniform stability to quasi-optimality and establishing solvability and error estimates under stated assumptions. It frames the error analysis of (μδ, uδ) by viewing system (3.15), before eliminating σδ, as a discretization of (2.8).
3.1. Uniformly stable (Petrov-) Galerkin discretizations and quasi-optimal approximations.
- Uniform stability is equivalent to quasi-optimal Petrov–Galerkin approximation, uniformly over the family of discrete spaces.Theorem 3.1 gives this equivalence for the stated Hilbert-space setting.Source: §3.1 · p. 5
3.2. Uniformly stable Galerkin discretizations of (2.10).
- For discretizations of (2.10), Theorem 3.3 establishes stability under its subspace hypotheses, with the stability constant growing unbounded as ∥Aa∥ increases.The Galerkin operator is invertible precisely when the Schur complement is invertible.Source: §3.2 · p. 7
3.3. Galerkin discretizations of (2.8).
- The result generalizes the inf-sup identity known for Yδ = Y to mesh-dependent norms; under (3.8), the theorem’s stated subspace conditions yield a unique Galerkin solution and a suboptimal error estimate.Source: §3.3 · pp. 9–10
- Without the additional condition (3.5), Theorem 3.5 does not establish quasi-best approximation, and examples show errors can exceed best-approximation errors.The text also notes numerical evidence that these approximations need not be quasi-optimal in the Y-norm.Source: §3.3 · pp. 10–11
- With (3.5) additionally assumed, Theorem 3.7 gives quasi-optimal error estimates for the Galerkin solutions of (2.8).This reproduces results from [And13, Thms. 3.1 & 4.1] with a clear-cut constant.Source: §3.3 · p. 12
- For (2.10), the operator and its inverse are uniformly bounded in the mesh-dependent norm, but the authors could not establish a suboptimal error estimate like Theorem 3.5.For (2.8), Remark 3.6 generalizes Steinbach’s result [Ste15, Theorem 3.2] and avoids forming normal equations.Source: §3.3 · pp. 10, 13
- Viewing (3.15) before eliminating σδ as a Galerkin discretization of (2.8) yields the sharp error bound for (μδ, uδ) in Theorem 3.7.Source: §3.3 · p. 14
4. REALIZATION OF THE UNIFORM INF-SUP STABILITY (3.5)
The section establishes conditions under which prismatic space-time finite element discretizations satisfy uniform stability. It covers both spatial spaces that vary between time slabs and tensor-product spaces with general temporal trial spaces.
4. REALIZATION OF THE UNIFORM INF-SUP STABILITY (3.5)
- Prismatic finite element spaces satisfy the stability condition when the spatial projections are uniformly V-stable and the stated space-time mesh conditions hold.The section treats non-uniform partitions whose time partition is independent of spatial location.Source: §4 · p. 14
4.1. Non-uniform approximation in space local in time, non-uniform approximation in time global in space.
- For spatial trial spaces varying between time slabs, uniform V-stability of their H-orthogonal projections is the key spatial condition.The text notes this property for quasi-uniform meshes and for locally refined meshes, including some generated by newest vertex bisection.Source: §4.1 · pp. 14–15
- When the spatial trial space is independent of the time interval, tensor-product spaces are inf-sup stable for general temporal trial spaces, including globally smooth spline spaces.The theorem takes Xδ = Xt ⊗ Xx and Yδ = Yt ⊗ Xx, with d/dt Xt ⊆ Yt.Source: §4.1 · pp. 15–16
- Sparse tensor-product approximations retain uniform inf-sup stability with constant µO.The result is stated uniformly in the sparse-level index.Source: §4.1 · p. 16
- For continuous piecewise-linear trial functions on a partition of I and continuous piecewise-linear tests on its once dyadically refined partition, an easy computation establishes an inf-sup lower bound.Source: §4.1 · p. 16
5. NUMERICAL EXPERIMENTS
The experiments compare the new mixed method with Andreev’s method on smooth and singular heat-equation solutions, then test a nonsymmetric convection-diffusion operator. Their errors show comparable convergence, while the new method’s system matrix has asymptotically about half as many nonzeros.
5. NUMERICAL EXPERIMENTS
- The new method’s whole-system matrix has asymptotically a factor of 2 fewer nonzeros than Andreev’s method.The paper presents this as a lower computational cost for the same accuracy.Source: §5 · p. 17
- The two methods’ errors are nearly indistinguishable for smooth and singular solutions, with observed convergence rates of 1/2 and 1/4, respectively.These rates are described as best possible given the polynomial degrees and solution regularity; Figure 1 plots X-norm error against dim Xδ for both solutions.Source: §5 · p. 17, Figure 1 · p. 17
- For both solutions and methods, the final-time L2(Ω) error converges at rate 1, asymptotically proportional to h_t^2.Figure 2 also plots the initial-time L2(Ω) error against dim Xδ.Source: §5 · p. 17, Figure 2 · p. 18
- For the symmetric problem, the secondary approximation’s Y-norm error is slightly larger than the primal approximation’s; the nonsymmetric case shows comparable X-norm convergence for both methods.The nonsymmetric test uses a convection-diffusion operator with β = 100 and the singular solution.Source: §5 · p. 18, Figure 3 · p. 18
6. CONCLUSION
The Steinbach scheme is cheapest and works on general space-time meshes, but can be far from quasi-optimal in the natural mesh-independent norm. Andreev’s method and the new mixed formulation give quasi-optimal approximations for suitable spaces, with the new method achieving the same accuracy at lower computational cost than Andreev’s.
6. CONCLUSION
- The Steinbach scheme has the lowest computational cost and applies on general space-time meshes, but its solutions can be far from quasi-optimal in the natural mesh-independent norm.Source: §6 · pp. 18–19
- For suitable trial-space pairs, the new mixed formulation gives quasi-optimal approximations at the same accuracy as Andreev’s method but at lower computational cost.Source: §6 · p. 19
- Andreev’s minimal-residual Petrov-Galerkin method gives quasi-optimal approximations from the trial space for suitable trial and test pairs.Source: §6 · p. 19