PaperlayerView exact revision on arXiv

Source-linked AI summary

A stabilized scheme satisfying the discrete maximum principle for a time fractional convection-diffusion-reaction equation

Christos Pervolianakis

arXiv:2609.03808v1math.NA

TL;DR

The paper addresses time-fractional convection–diffusion–reaction equations where nonsmooth initial data and layers motivate accurate, maximum-principle-preserving discretizations. It combines conforming linear finite elements with algebraic flux correction, analyzes the resulting nonlinear schemes, and obtains optimal convergence behavior with discrete maximum-principle guarantees. The fully discrete L1 scheme extends these guarantees in time, while the authors identify a mesh-assumption boundary for optimal L2 error estimates.

  • Problem

    Time-fractional problems can have initial layers and convection–diffusion–reaction solutions can contain steep spatial layers, motivating numerical schemes that preserve the discrete maximum principle for nonsmooth data.

  • Method

    The paper combines conforming linear finite elements with algebraic flux correction, uses fixed-point arguments for nonlinear scheme well-posedness, and applies L1 discretization on a uniform temporal mesh.

  • Results

    The stabilized schemes satisfy the discrete maximum principle, while the semi-discrete analysis derives optimal h-convergence rates under 0 < α < 1/(2−δ).

  • Takeaways & Limitations

    The proposed AFC stabilization provides a maximum-principle-preserving finite element approach whose reported numerical results retain the optimal convergence order for layer-containing solutions.

  • Takeaways & Limitations

    Optimal L2 error estimates in h are unavailable without the acute-mesh assumption because the relevant matrix-entry estimate can become O(1) in the diffusion-dominant regime.

Abstract

from arXiv · show

We study a time fractional convection-diffusion-reaction equation in a bounded domain $Ω\subset\mathbb{R}^2$. A stabilized numerical scheme satisfying a discrete maximum principle is constructed by combining the conforming linear finite element method with the algebraic flux correction method. The resulting semi-discrete scheme is nonlinear, and its well-posedness is established. Assuming nonsmooth initial data, we derive error estimates for the semi-discrete scheme using energy arguments. For the temporal discretization, we employ the L1 method, obtaining a fully discrete scheme for which we prove well-posedness and the discrete maximum principle. We also present numerical experiments that validate the order of convergence as well as we test our schemes to solutions that possess layers.

1. Introduction

The paper develops stabilized finite element schemes for time-fractional convection–diffusion–reaction equations, targeting discrete maximum-principle preservation despite initial layers and spatial solution layers. It combines conforming linear finite elements with algebraic flux correction, establishes well-posedness and error estimates, and uses L1 time discretization for a fully discrete scheme.

  • 1. Introduction: Nonsmooth initial data create an initial layer at t = 0, while convection–diffusion–reaction solutions can develop narrow regions with steep gradients.These features can cause time-derivative singularities and spurious finite-element oscillations.
  • 1.2. Finite element method.: The proposed spatial discretization combines conforming linear finite elements with algebraic flux correction to enforce a discrete maximum principle.The AFC stabilization makes the semi-discrete scheme nonlinear.
  • 1.3. Contributions and outline of the paper.: The nonlinear semi-discrete scheme has established existence and uniqueness for sufficiently small mesh size h, together with discrete maximum-principle preservation.The well-posedness proof uses a fixed-point argument.
  • 1.3. Contributions and outline of the paper.: For nonsmooth initial data, an energy-argument analysis treats the nonlinear stabilization term and derives optimal h-convergence rates when 0 < α < 1/(2−δ).The analysis is designed for initial data with only L2 regularity rather than H2 regularity.
  • 1.3. Contributions and outline of the paper.: The fully discrete method uses L1 discretization on a uniform temporal mesh, with fixed-point well-posedness and local and global discrete maximum-principle results.The paper also reports numerical experiments for layer-containing solutions and convergence-order validation.

2. Preliminaries

The preliminaries introduce fractional integral and derivative operators, energy-estimate tools, and regular triangulation assumptions. The mesh assumptions include shape regularity, quasi-uniformity, and an acute-angle condition supporting matrix sign properties.

  • 2. Preliminaries: The analysis recalls the Riemann–Liouville fractional integral and Caputo fractional derivative, together with their semigroup and continuity properties.These operator identities and inequalities are used in the subsequent fractional energy analysis.
  • 2. Preliminaries: An energy-estimate inequality for absolutely continuous functions is introduced as a key tool for the analysis.The paper identifies this inequality as central to its energy estimates.
  • 2.1. Mesh assumptions.: The finite element meshes are required to be regular, shape-regular, and quasi-uniform, with geometric constants independent of the mesh.These assumptions provide the stated mesh-independent geometric framework.
  • 2.1. Mesh assumptions.: The acute-mesh condition requires every triangle angle to be at most π/2 and implies nonpositive off-diagonal stiffness entries.This sign property is later identified as important for the stabilization analysis.
  • 2. Preliminaries: For each node Zi, the patch ωi is the union of triangles sharing that vertex, with neighboring nodes and local vertices or edges defined on the patch.Figure 2.1 depicts this nodal subdomain and its eight neighboring nodes.

3. Algebraic flux correction

The scheme combines conforming linear finite elements with algebraic flux correction to enforce the discrete maximum principle, while retaining nonlinear stabilization and provable well-posedness.

  • A semi-discrete scheme that satisfies the DMP: The algebraic flux correction replaces the mass matrix by a lumped mass matrix and adds symmetric artificial diffusion to control negative off-diagonal entries.Anti-diffusive fluxes are multiplied by solution-dependent correction factors before insertion into the scheme.
  • A semi-discrete scheme that satisfies the DMP: The resulting variational formulation is nonlinear because the stabilization terms and correction factors depend on the discrete solution.The coefficient form includes the mass, diffusion, convection, reaction, and correction terms.
  • Correction factors: The correction factors are constrained through local extrema and fluxes, with symmetry, boundedness in [0, 1], linearity preservation, and a local estimate.The algorithm computes pairwise factors and symmetrizes them across neighboring nodes.
  • Well-posedness of AFC semi-discrete scheme: For sufficiently small h, the nonlinear stabilized semi-discrete scheme has a unique solution because its fixed-point mapping is contractive.The contraction condition is stated as Cµh^2 < 1.
  • Discrete Maximum Principle: The nonlinear stabilized semi-discrete scheme satisfies the global discrete maximum principle under the stated correction-factor construction.The result applies to correction factors defined through the specified limiter and algorithm.

4. Error analysis

The error analysis uses elliptic projection and energy arguments to estimate the stabilized semi-discrete error for nonsmooth initial data, under regularity and fractional-order restrictions.

  • Error analysis: The analysis splits the error through an elliptic projection and applies time integration, fractional identities, inverse inequalities, and Gronwall arguments.The discrete error satisfies an error equation whose stabilization contributions are estimated separately.
  • Error analysis: The nonlinear stabilization term is treated through estimates involving the correction factors and the auxiliary bounds for the stabilized bilinear form.These estimates support the energy argument for nonsmooth initial data.
  • Error analysis: Theorem 4.1 derives an L2 error estimate for nonsmooth initial data under the regularity assumptions and for 0 < α < 1/(2−δ).The estimate concerns the unique exact and stabilized semi-discrete solutions and uses a constant independent of h.
  • Error analysis: The proof requires integrability conditions on fractional-time terms, including 0 < α < 1/(2−δ) for one Gronwall estimate.The restriction arises from finiteness of the relevant fractional integral.

5. A fully–discrete scheme that satisfies the DMP

The fully discrete method combines uniform-mesh L1 time discretization with nonlinear AFC stabilization, and it is proved uniquely solvable while satisfying local and global discrete maximum principles.

  • 5.1. Temporal discretization and well-posedness: The L1 method on a uniform temporal mesh produces a nonlinear fully discrete scheme whose existence and uniqueness are established.The nonlinearity comes from the stabilization terms, and the proof uses a fixed-point argument.
  • 5.1. Temporal discretization and well-posedness: The L1 weights decrease with their index, while the standard truncation-error estimate requires sufficient solution regularity.For twice continuously differentiable solutions, the local truncation error is bounded by c1k^(2−α).
  • 5.1. Temporal discretization and well-posedness: The fully discrete scheme is formulated by approximating each time level U n in the finite element space and testing against all χ in that space.
  • 5.1. Temporal discretization and well-posedness: A contractive operator argument for sufficiently small h establishes a unique solution at each time step.The contraction follows after choosing h so that Cµh < 1.
  • 5.2. Discrete maximum principles: The stabilized fully discrete scheme satisfies the local discrete maximum principle for all time-step and mesh sizes k and h.
  • 5.2. Discrete maximum principles: The stabilized fully discrete scheme also satisfies the global discrete maximum principle for all k and h.The result is obtained under the correction-factor assumptions used for the local principle.

6. Numerical experiments

Numerical experiments compare standard FEM and AFC on nonsmooth and layered solutions. They show that AFC suppresses spurious oscillations and achieves optimal experimental convergence orders in the tested norms and parameter regimes.

  • 6. Numerical setup: The experiments use uniform right-triangular meshes, uniform time-step refinement, fixed-point iteration, and Aitken acceleration with tolerance 10−5.The fixed-point procedure is capped at 50 iterations.
  • 6.1. A numerical example with nonsmooth initial data: AFC produces a non-oscillatory profile, whereas standard FEM is dominated by spurious oscillations for the nonsmooth-initial-data test.The comparison uses M = 40 and N0 = 100, with µ = 10−12, α = 0.2, and a strong convection field.
  • 6.2. Convergence study with solution with steep gradients: AFC achieves the optimal experimental order of convergence in both L2 and energy norms for steep-gradient solutions with µ = 10−3, 10−4, and 10−5.Results for α = 0.2 and α = 0.5 are reported as very close.
  • 6.3. Convergence study with solution with circular interior layer: AFC achieves the optimal experimental order in both L2 and energy norms for the circular interior-layer test across µ = 10−3, 10−4, and 10−5.The result is reported for both α = 0.2 and α = 0.5.
  • 6.4. Convergence study with solution with interior layer: AFC achieves the optimal experimental order in both L2 and energy norms for the interior-layer test along a line.The layer has width O(√µ), and the experiments use µ = 10−3, 10−4, and 10−5.

7. Conclusions

The paper concludes that AFC stabilization yields a discrete-maximum-principle-preserving method for time fractional convection–diffusion–reaction equations, with optimal convergence demonstrated on layered problems.

  • 7. Conclusions: The proposed stabilized finite element scheme satisfies the discrete maximum principle for the considered time fractional convection–diffusion–reaction problem.
  • 7. Conclusions: For nonsmooth initial data, the paper derives L2 error estimates for the stabilized semidiscrete scheme under a restriction on α.
  • 7. Conclusions: Numerical experiments involving solutions with layers demonstrate that the proposed stabilized scheme preserves the optimal convergence order.
Loading 2609.03808v1…