A space-time sparse-grid method for the wave equation

Abstract: We develop a fast space-time numerical scheme for approximating solutions to the linear wave equation. The approach is based on the sparse-grid combination technique applied to a coercive space-time discretization. Designed for tensor-product space-time discretizations, the method enables efficient parallelization of the resulting solver. We provide a rigorous theoretical analysis establishing convergence rates and computational complexity estimates. Numerical experiments validate the theoretical estimates and demonstrate the efficiency of the proposed method.

• The paper develops a coercive conforming space–time Galerkin method combined with a sparse-grid combination technique, supporting arbitrary tensor-product meshes and independent parallel solves.
• The paper proves optimal-order L² convergence, with error rates of 2⁻ᴶᵐⁱⁿ⁽ᵖˣ⁺¹,ᵖᵗ⁺¹⁾ and a logarithmic factor when spatial and temporal degrees match, under mixed regularity and compatibility assumptions.
• The paper shows that sparse grids reduce complexity from roughly 2⁽ᵈ⁺¹⁾ᴶ to 2ᵈᴶ degrees of freedom in dimensions 2 and 3, while experiments confirm improved accuracy per degree of freedom but reveal sensitivity to coarse-grid choice and compatibility conditions.

Overview

This paper develops and analyzes a fast space–time numerical scheme for the linear wave equation, combining a coercive conforming space–time Galerkin discretization with the sparse-grid combination technique. The underlying discretization is a second-order-in-time variational formulation in which integration by parts is performed in space but not in time; an initial-condition penalty term weakly enforces $\partial_t u(\cdot,0)=v_0$, and exponentially weighted $L^2$ scalar products with weight $e^{-t/T}$ guarantee coercivity of the bilinear form in the energy norm $H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$ for arbitrary tensor-product discrete spaces with $C^1$ temporal component. Building on this unconditional stability, the authors prove optimal-order convergence of the combination-technique solution directly — not of a Galerkin solution on the sparse-grid space — for arbitrary polynomial degrees in space and time.

The underlying space–time Galerkin scheme

The model problem is the initial-boundary value problem for $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$ on the space–time cylinder $Q_T = \Omega\times(0,T)$, with homogeneous Dirichlet conditions. The trial/test space is

$$V(Q_T) = (H_0^1(\Omega)\otimes H^1(0,T)) \cap (L^2(\Omega)\otimes H^2_{0,\bullet}(0,T)),$$

and the bilinear form involves $(\partial_t^2 u,\partial_t w)$, an initial-time penalty term, and $(c^2\nabla_x u,\nabla_x\partial_t w)$, all weighted by $L^2$0. A key algebraic identity shows that the exponential weight makes the cross term $L^2$1 controllable, yielding coercivity

$L^2$2

Continuity does not hold with respect to the same norm, so quasi-optimality is not automatic for arbitrary subspaces; optimal rates are known only for specific tensor-product choices, e.g., $L^2$3 splines of even degree in time on uniform meshes. This limitation matters here because the combination technique requires solving problems on strongly anisotropic meshes, where stability must hold unconditionally.

Two tailored projection operators underpin the analysis: a spatial Ritz-type projection $L^2$4 associated with the operator $L^2$5, and a non-standard temporal projection $L^2$6 defined via second time derivatives. Their approximation properties (orders $L^2$7 and $L^2$8 in $L^2$9, respectively) yield the full-grid error estimate

$e^{-t/T}$0

under mixed regularity of the solution. The validity of the temporal approximation property is proven for $e^{-t/T}$1 splines of even degree and conjectured for other spline regularity/degree parities — a gap the paper states explicitly.

The combination technique and complexity

For maximal level $e^{-t/T}$2, the combination formula assembles the sparse-grid approximation from $e^{-t/T}$3 Galerkin solutions on diagonal spaces $e^{-t/T}$4:

$e^{-t/T}$5

The degrees-of-freedom count is decisive: while the full grid scales as $e^{-t/T}$6, the sparse grid satisfies $e^{-t/T}$7 for $e^{-t/T}$8 and $e^{-t/T}$9 for $H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$0. Since each of the $H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$1 systems can be solved independently, the method is embarrassingly parallel — the paper positions this as one of the few conforming space–time wave solvers that combines parallelizability with a rigorous convergence analysis, contrasting it with fast-diagonalization approaches whose conforming stability analysis remains open, and with tent-pitching DG schemes which are non-conforming.

Main convergence result

The central theorem establishes, under domain/coefficient regularity ($H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$2, $H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$3), mixed regularity

$H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$4

and discrete compatibility conditions at $H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$5 and on $H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$6, the estimate

$H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$7

In terms of degrees of freedom this gives $H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$8 versus $H^1(0,T;L^2(\Omega))\cap L^2(0,T;H^1_0(\Omega))$9 for the full grid — asymptotically a strictly better rate per DoF for all $C^1$0.

The proof strategy is notable: rather than analyzing a Galerkin method posed on the sparse-grid space (which, given the lack of norm-coincident continuity, could yield reduced rates), the authors bound the difference between the combination solution and the full-grid projection by summing detail operators. They introduce a "detail projection" $C^1$1 built from the tensor-product projections $C^1$2, whose decay follows directly from approximation properties, and control the gap $C^1$3 via semidiscretization operators $C^1$4 and $C^1$5. The required stability estimates (spectral-norm bounds for $C^1$6, energy-type bounds for $C^1$7 with explicit initial-time recursions involving powers of the discrete operator $C^1$8) constitute the technical core, and they hold precisely when the discrete compatibility conditions are satisfied.

Role of the compatibility conditions

The discrete compatibility conditions require that certain time derivatives of $C^1$9 at $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$0 lie in the spatial discrete space, and that $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$1 vanish appropriately on $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$2; equivalently, combinations of derivatives of $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$3 must belong to the discrete spaces. The paper is candid about their status:

• They are implied by stronger, easily checkable conditions on the data (e.g., vanishing traces of $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$4 on the boundary).
• Numerical evidence on detail operators shows suboptimal decay of $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$5 when the conditions fail, with the optimal rate recovered as conditions are progressively enforced.
• Conversely, some experiments (a smooth solution on $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$6 with $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$7) achieve optimal rates despite violating all compatibility conditions, indicating the assumptions are sufficient but not necessary.

A $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$8-dimensional test confirms that ignoring them entirely can cause order reduction: for one solution both methods converge optimally at $\partial_t^2 u - \nabla_x\cdot(c^2\nabla_x u) = f$9, but the sparse-grid rate degrades at $Q_T = \Omega\times(0,T)$0 when initial conditions fail. Identifying minimal sufficient conditions remains open.

Numerical validation

Experiments in $Q_T = \Omega\times(0,T)$1 and $Q_T = \Omega\times(0,T)$2 dimensions, using $Q_T = \Omega\times(0,T)$3/maximal-regularity splines and simplicial finite elements, confirm the predicted $Q_T = \Omega\times(0,T)$4 convergence for both full- and sparse-grid solutions, and the improved DoF-to-error behavior of the sparse grid, matching the predicted rates including logarithmic factors. Notably, optimal rates are observed even with maximal-regularity splines in time, for which the key approximation assumption is only conjectured — evidence supporting that conjecture but not proving it. A traveling-wave experiment shows the relative energy error remains uniformly bounded over $Q_T = \Omega\times(0,T)$5 for both schemes, indicating no energy drift.

One experiment exposes a practical sensitivity: with fixed finest mesh, sparse-grid accuracy depends strongly on the coarsest mesh size $Q_T = \Omega\times(0,T)$6. With $Q_T = \Omega\times(0,T)$7 the expected rate fails entirely for $Q_T = \Omega\times(0,T)$8; refining $Q_T = \Omega\times(0,T)$9 progressively restores convergence. The paper leaves open the question of criteria for choosing $$V(Q_T) = (H_0^1(\Omega)\otimes H^1(0,T)) \cap (L^2(\Omega)\otimes H^2_{0,\bullet}(0,T)),$$0 that guarantee onset of the asymptotic regime — a genuine pre-asymptotic effect inherent to the combination formula.

Limitations and open questions

Several restrictions qualify the results. The convergence theory requires mixed regularity of order up to $$V(Q_T) = (H_0^1(\Omega)\otimes H^1(0,T)) \cap (L^2(\Omega)\otimes H^2_{0,\bullet}(0,T)),$$1, attainable only for sufficiently smooth data satisfying high-order compatibility; whether these hypotheses are sharp is unresolved. The temporal approximation property is proven only for $$V(Q_T) = (H_0^1(\Omega)\otimes H^1(0,T)) \cap (L^2(\Omega)\otimes H^2_{0,\bullet}(0,T)),$$2 splines of even degree, with other cases conjectural. The discrete compatibility conditions cannot be dispensed with in general, yet their minimal form is unknown. The analysis targets the $$V(Q_T) = (H_0^1(\Omega)\otimes H^1(0,T)) \cap (L^2(\Omega)\otimes H^2_{0,\bullet}(0,T)),$$3 norm only; energy-norm estimates for the combination solution are not provided. Finally, the extension of the combination technique to the spatial dimensions (for tensor-product domains) and quantitative pre-asymptotic criteria for $$V(Q_T) = (H_0^1(\Omega)\otimes H^1(0,T)) \cap (L^2(\Omega)\otimes H^2_{0,\bullet}(0,T)),$$4 are identified as directions not settled by this work.

Conclusion

The paper provides a complete convergence theory for a sparse-grid combination technique applied to an unconditionally stable, coercive conforming space–time Galerkin method for the wave equation. It achieves optimal-order $$V(Q_T) = (H_0^1(\Omega)\otimes H^1(0,T)) \cap (L^2(\Omega)\otimes H^2_{0,\bullet}(0,T)),$$5 accuracy with $$V(Q_T) = (H_0^1(\Omega)\otimes H^1(0,T)) \cap (L^2(\Omega)\otimes H^2_{0,\bullet}(0,T)),$$6 degrees of freedom instead of $$V(Q_T) = (H_0^1(\Omega)\otimes H^1(0,T)) \cap (L^2(\Omega)\otimes H^2_{0,\bullet}(0,T)),$$7, natural parallelism across independent diagonal solves, and numerically validated rates — at the cost of mixed-regularity and discrete compatibility assumptions whose necessity is only partially characterized.