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Finite Volume Convergence for a Heat Equation with a Measure-Valued Source
Matthias Gobbert, Martin Kruž\'ık, Andreas Meister, Michael Muscedere
TL;DR
Parabolic finite-volume convergence theory for genuinely measure-valued sources is less explicit than for L1 data. This paper extends prior finite-element analysis to the classical two-point flux method and finds dimension-dependent rates, with computations exhibiting the predicted behavior.
Problem
Finite-volume convergence theory is well developed for parabolic L1 data, but is less explicit for heat equations with genuinely Radon-measure right-hand sides.
Source: §1 · p. 3Method
The paper analyzes the classical two-point flux finite-volume method, extending the finite-element approach through stationary and time-dependent Dirac-source problems.
Source: §1 · pp. 2–3Results
The predicted convergence orders are 1 in two dimensions and 1/2 in three dimensions, and computational tests exhibit this behavior.
Source: §1 · pp. 3–4Takeaways & Limitations
The computational evidence confirms the analytical behavior, and the authors identify possible extension to methods such as discontinuous Galerkin.
Source: §1 · p. 4The analysis sets D = 1 and assumes quasi-uniform meshes; available estimates on general admissible meshes are generally first order.
Source: §2.2 · pp. 7–8
Abstract
from arXiv · showhide
Past work with Tom Seidman analyzed the spatial convergence order of the numerical finite element method applied to time-dependent diffusion-reaction equations with a measure-valued source term. For smooth source terms, the numerical method has second-order accuracy, but the non-smooth source terms introduce a limitation of the order that depends on the spatial dimension. This work in honor of Tom extends the analysis to the finite volume method in its clas- sical two-point flux form and shows that the regularity of the source term reduces the convergence order to 1 in two and 0.5 in three spatial dimensions which is reduced from the nominal second-order and depends on the space dimension. Computational tests with a point source modeled by a Dirac delta distribution exhibit the behavior predicted by the analytical results.
1 Introduction
The paper extends finite-element convergence analysis for measure-valued sources to the classical two-point flux finite-volume method, addressing a gap in parabolic finite-volume theory for Radon-measure data. It proves dimension-dependent convergence rates and reports computational tests consistent with them.
1 Introduction
- The paper extends the finite-element result in to the classical two-point flux finite-volume method, with essentially the same dimension-dependent behavior.The analysis proceeds from a stationary problem to heat equations with space-time and persistent Dirac sources.Source: §1 · pp. 2–3
- Finite-volume convergence theory is established for parabolic L1 data, while direct results for genuinely Radon-measure data appear less explicit in the standard parabolic literature.The corresponding heat-equation problem is understood at the continuous PDE level.Source: §1 · p. 3
- Finite-volume errors for point sources converge at order 1 in two dimensions and 1/2 in three dimensions, as predicted by the analysis.Computational tables exhibit these rates for stationary and time-dependent problems.Source: §1 · pp. 2, 4
- The analysis is accompanied by Matlab tests in two and three dimensions, and the authors note potential extension to methods such as discontinuous Galerkin.The computational evidence is reported to confirm the analytical behavior.Source: §1 · p. 4
2 Finite Volume Method Discretization
The method integrates the diffusion-reaction equation over control volumes and approximates diffusive fluxes with the classical two-point flux scheme. The analysis uses cell-average notation and distinguishes smooth-solution accuracy from the behavior with singular sources.
2.1 General formulation
- Integrating over control volumes converts the diffusion term to surface fluxes, enabling discrete mass conservation and finite-volume treatment of a Dirac point source.The method uses control-volume source integrals and does not distinguish a Dirac source from its uniform distribution within the containing cell.Source: §2.1 · pp. 4–5
- Each control volume carries one cell-average unknown, and each interior-face flux uses a difference quotient between neighboring cell values.The fluxes are conservative, and each unknown interacts only with face neighbors without higher-order reconstruction.Source: §2.1 · p. 5
- The semi-discrete system uses cell-average unknowns and source integrals, with lumped mass and stiffness matrices that exploit sparse Kronecker-product structure.Time-dependent computations use implicit Euler stepping and conjugate gradients; stationary systems use sparse direct Gaussian elimination.Source: §2.1 · pp. 5–6
2.2 Notation and accuracy of the scheme used in the analysis
- The analysis assumes quasi-uniform meshes and identifies the cell containing the source; a discrete maximum principle follows from the symmetric M-matrix structure.For a Dirac source, the discrete source is uniform over the source-containing cell.Source: §2.2 · p. 7
- For smooth solutions on the uniform meshes, the scheme is second-order accurate in the discrete L2 norm and at least first-order in the discrete energy norm.On general admissible meshes, available two-point-flux error estimates are generally first order; the piecewise-constant L2 error also includes an O(h) projection error.Source: §2.2 · pp. 7–8
3 Finite Volume Error for a Poisson Equation with a Dirac Source
For a Dirac source, the two-point flux finite volume approximation has dimension-dependent error behavior: local singularity errors and their propagation through the discrete solution determine the global convergence rate. The section develops heuristic scaling arguments for two and three dimensions, then gives a rigorous bound under a specific mesh and scheme assumption.
3.1 The two-dimensional case
- In 2-D, the rigorous error is O(h|log h|^1/2), as local flux errors and piecewise-constant cell approximation affect the heuristic far-field estimate.The discrete solution reproduces the logarithmic growth on the singular cell to leading order, while the rigorous analysis accounts for global error propagation.Source: §3.1 · p. 10
3.2 The three-dimensional case
- In 3-D, the singular cell alone contributes an error of order h^1/2, which is also a lower bound because the finite volume solution is constant on that cell.The heuristic scaling argument suggests this order, and the rigorous proof accounts for propagation of local errors.Source: §3.2 · pp. 11–12
3.3 Rigorous error bound including the global propagation of local errors
- Under Assumption (A), the rigorous bound is O(h|log h|^1/2) in 2-D and O(h^1/2) in 3-D; in 3-D the singular cell also gives a matching lower bound.The proof accounts for propagation of local flux errors through the globally coupled discrete problem.Source: §3.1 · p. 10, §3.2 · p. 11, §3.3 · pp. 14–15
- The rigorous argument identifies finitely many source-adjacent faces with order-one flux errors; their propagation through the discrete dual solution contributes to the global error.For general quasi-uniform admissible meshes, the analysis states the same rates up to an additional logarithmic factor if a dual stability estimate is available.Source: §3.3 · p. 15
- With an additional discrete W^1,∞ stability estimate, the duality argument yields an O(h) L1 error bound in both 2-D and 3-D.The passage also states that the projection error is O(h) when ∇u is integrable.Source: §3.3 · p. 16
4 Finite Volume Error for a Heat Equation with a Space-Time Dirac Source
For a space-time Dirac source, the paper analyzes the two-point flux finite volume method after the impulse and derives dimension-specific error bounds. In two dimensions it recovers the standard spatial rates; in three dimensions it gives a corresponding fixed-time bound.
4.1 Exact solution in the whole space
- The impulse creates an instantaneous singularity: the solution is measure-valued at t = t0 but smooth in space at every fixed later time.Source: §4.1 · p. 16
4.2 Scaling of the solution
- Heat-kernel self-similarity and its spatial derivatives provide the scaling framework for estimating the singular solution.The derivatives are represented by a polynomial times a Gaussian, and the analysis takes their L2 norm after a change of variables.Source: §4.2 · p. 17
4.3 Finite volume approximation
- For smooth solutions, the low-order two-point flux scheme is second-order accurate in the discrete L2 norm and first-order in the discrete energy norm.The analysis applies these smooth-solution estimates at each fixed time after the impulse, when the heat kernel is smooth.Source: §4.3 · pp. 17–18
- The general L2 error estimate scales as h2 s^-1-d/4 for s = t − t0 > 0.Here s is the elapsed time after the impulse.Source: §4.3 · p. 18
4.4 Explicit rates in two dimensions
- In 2-D, the method attains O(h2) in L2 and O(h) in the energy norm for every fixed positive time after the impulse.The explicit error bound is ∥u(·, t) − uh(·, t)∥L2(Ω) ≲ h2 s^-3/2, where s = t − t0 > 0.Source: §4.4 · p. 18
4.5 Explicit rates in three dimensions
- For the low-order two-point flux finite volume method, for every fixed t > t0, the L2 error ∥u(·, t) − uh(·, t)∥L2(Ω) is ≲ h2 s−7/4, with s = t − t0 > 0.Source: §4.5 · pp. 18–19
5 Finite Volume Error for a Heat Equation with a Persistent Dirac Source
For a persistent Dirac source, the analysis controls finite volume errors after a positive time delay by regularizing the source and separating source, spatial, and time-discretization errors. It also gives backward Euler error bounds in two and three dimensions.
5 Finite Volume Error for a Heat Equation with a Persistent Dirac Source
- The proof avoids consistency estimates involving derivatives near the source cell, whose higher derivatives grow as h decreases.It instead uses a stationary problem and semigroup estimates to control the finite volume error.Source: §5 · p. 21
- After a positive delay, combining the stationary, smoothing, and discrete-error estimates gives an L2 error bounded by Chℓh.The same analysis bounds the L1 error; the discrete semigroup is contractive in the discrete L1 norm.Source: §5 · p. 21
- On compact subsets away from x0, finite volume consistency yields optimal local convergence rates.Source: §5 · p. 22
- For tn ≥ t0 + τ, the backward Euler error is bounded by C(h|log h|^1/2 + ∆t) in 2-D and C(h^1/2 + ∆t) in 3-D.Source: §5 · p. 22
6 Stationary and Time-Dependent Numerical Tests
The numerical tests examine stationary and transient finite-volume diffusion-reaction problems with a Dirac point source; both show dimension-dependent spatial convergence consistent with analysis.
6.1 Stationary diffusion-reaction equation with homogeneous Dirichlet boundary conditions
- The stationary test models pollutant diffusion on (-50, 50)^d with D=40 and a central Dirac source injected at rate κ=10.Because the true solution is unavailable, errors are measured against the finest available numerical reference solution.Source: §6.1 · p. 23
6 Stationary and Time-Dependent Numerical Tests
- Observed orders approach 1 in 2-D and 0.5 in 3-D for both tests, with transient results consistent across t=4, 8, and 12.The estimated orders are judged from coarser refinement levels because reference-solution self-comparison can make the finest-level order appear too large.Source: Table 1 · p. 24, §6.1 · p. 24, Table 3 · p. 26
6.1 Stationary diffusion-reaction equation with homogeneous Dirichlet boundary conditions
- Finest-level stationary error estimates can be too small because the numerical solution and reference solution become increasingly correlated.The authors recommend judging estimated orders from coarser refinement levels.Source: §6.1 · p. 24
6.2 Time-dependent diffusion-reaction equation with homogeneous Dirichlet boundary conditions
- The transient test uses a central point source injected for 24 hours, with implicit Euler Δt=1/128 and tight CG tolerance 10^-12 to isolate spatial convergence.The tests report errors at t=4, 8, and 12.Source: §6.2 · p. 25