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Finite Element Approximation of Rotating MHD-Boussinesq Flows with Nonlinear Power-Law Viscosity

Akash Ashirbad Panda, Evana Islam Sarkar

arXiv:2610.31091v1math.NAcs.NA

TL;DR

The paper addresses the lack of a fully discrete finite element analysis for rotating MHD-Boussinesq flows with power-law viscosity. It analyzes a semi-implicit finite element scheme that preserves key energy properties. The scheme is well posed and energy stable without mesh-time coupling, and the Newtonian case has the classical error rate.

  • Problem

    A fully discrete finite element analysis of the three-dimensional rotating MHD-Boussinesq system with an H(curl)-conforming magnetic approximation had not been established in the cited literature.

    Source: §1.2 · p. 3

  • Method

    The method combines conforming finite elements for velocity and temperature, H(curl)-conforming Nédélec elements for magnetic induction, and semi-implicit backward Euler time stepping.

    Source: §1.1 · p. 2, §8 · p. 39

  • Results

    O(∆t + h^l) for p = 2; the scheme is well posed for arbitrary h and ∆t and has mesh-independent energy stability without mesh-time coupling.

    Source: §8 · pp. 39–40

  • Takeaways & Limitations

    Compactness and passage to the limit hold for p ≥ 2, while identification of the nonlinear flux is obtained in the Newtonian case and is restricted in three dimensions by the Lorentz force.

    Source: These are compatible whenever · p. 14, §8 · p. 40

    Flux identification outside the stated range appears to require additional tools, such as Lipschitz-truncation techniques.

    Source: These are compatible whenever · p. 14, §8 · p. 40

Abstract

from arXiv · show

We propose and analyse a fully discrete finite element approximation of the three-dimensional incompressible rotating MHD-Boussinesq system with nonlinear power-law ($p$-Laplace) viscosity. Conforming elements are used for the velocity and temperature, $\mathbf H(\operatorname{curl})$-conforming N\'ed\'elec elements for the magnetic induction, and a semi-implicit backward Euler scheme in time. The discretisation preserves the principal energy structure: the Lorentz and induction terms cancel exactly, while the Coriolis force remains energy neutral. This yields well-posedness and mesh-independent energy stability. The nonlinear power-law diffusion creates two main analytical difficulties. Since a uniform time-derivative estimate for the discrete velocity is unavailable, strong compactness is obtained through a time-translate argument and Simon's compactness criterion. The weak limit of the nonlinear diffusion is then identified by a Minty argument based on a discrete energy $\limsup$ inequality. Compactness and passage to the limit hold throughout the shear-thickening regime, while identification of a genuine weak solution is obtained in the Newtonian case and, in three dimensions, above an explicit threshold dictated by the Lorentz force rather than by convection. We further establish a priori error estimates without any mesh-time coupling condition. Three-dimensional numerical experiments confirm the stability and convergence behaviour predicted by the analysis.

1. Introduction

The paper addresses the gap in fully discrete finite element analysis for rotating MHD-Boussinesq flows with power-law viscosity. It develops a structure-preserving scheme and analyzes its stability, convergence, error estimates, and numerical behavior.

1.2. State of the art

  • No fully discrete finite element analysis for this three-dimensional rotating MHD-Boussinesq combination with an H(curl)-conforming magnetic approximation had been established in the cited literature.The system combines nonlinear viscosity, magnetic induction, buoyancy-driven temperature evolution, and Coriolis forces.Source: §1.2 · p. 3

1.3. Contributions

  • The method combines conforming finite elements for velocity and temperature, Nédélec edge elements for magnetic induction, and semi-implicit backward Euler time stepping.Its discrete energy structure uses exact cancellation of the Lorentz and induction terms, while the Coriolis contribution is energy neutral.Source: §1.3 · p. 3
  • Because the p-Laplacian prevents a uniform discrete velocity time-derivative bound, compactness uses time translates and Simon’s criterion, followed by Minty identification of the nonlinear flux.Compactness and passage to the limit hold for p ≥ 2; in three dimensions, the restriction for flux identification is dictated by the Lorentz force rather than convection.Source: §1.3 · p. 3
  • Under the stated assumptions, the analysis establishes a priori error estimates in L∞(L2), reducing to the classical O(∆t + h^l) rate when p = 2.The estimates require a pointwise divergence-free velocity pair; the Coriolis contribution is controlled without changing the resulting rate.Source: §1.3 · p. 3
  • Three-dimensional manufactured-solution experiments illustrate energy stability, first-order temporal convergence, and expected spatial convergence in Newtonian and shear-thickening regimes.The introduction describes magnetoconvection as arising in applications including planetary dynamo action, liquid-metal cooling, and plasma confinement.Source: §1 · p. 1, §1.3 · p. 4

2. Functional Setting and Mathematical Formulation of the Problem

The section sets the function spaces and identities used to formulate and analyze the rotating MHD-Boussinesq system. It establishes the compactness and energy properties needed for the later discrete analysis.

2.1. Standing assumptions and notation

  • The analysis assumes a bounded, simply connected Lipschitz polyhedron with connected boundary and restricts the power-law exponent to p ≥ 2.The Newtonian case is p = 2, and p > 2 represents shear-thickening.Source: §2.1 · p. 4

2.2. Function spaces

  • The velocity space incorporates no-slip and divergence-free constraints, while the temperature space incorporates its homogeneous Dirichlet condition.The pressure space supports the discrete inf-sup condition, but the discrete scheme is posed directly on a divergence-free velocity space.Source: §2.2 · pp. 4–5
  • The magnetic space H(curl;D) imposes the normal condition essentially, leaves the tangential trace unprescribed, and defines a solenoidal space incorporating div b = 0 and n·b = 0.Under the stated domain topology, the curl norm is a norm on the solenoidal magnetic space.Source: §2.2 · pp. 4–5

2.3. Embedding, Poincaré, and compactness results

  • The curl-div embedding gives the magnetic field additional spatial integrability needed to control the Lorentz force.The section also collects Poincaré and Sobolev estimates for the bounded Lipschitz polyhedron.Source: §2.3 · p. 5
  • Because no suitable uniform bound on the discrete velocity time derivative is available, Simon’s translation criterion replaces the classical Aubin-Lions approach for velocity compactness.The method uses a time-translate bound in place of the time-derivative estimate.Source: §2.3 · pp. 5–6

2.4. The nonlinear and trilinear forms

  • The skew-symmetric convective forms vanish when tested with repeated arguments, and the Coriolis force is energy neutral without requiring additional velocity regularity.For p ≥ 2, the nonlinear diffusion operator is strongly monotone.Source: §2.4 · p. 6

2.5. Weak formulation

  • The Lorentz and induction terms cancel exactly in the continuous energy balance, and the weak formulation’s rotational term vanishes when tested with velocity.The weak solution equations hold in the sense of distributions in time, with brackets denoting appropriate duality pairings.Source: §2.4 · p. 6, §2.5 · p. 7

3. The Fully Discrete Scheme

The scheme combines conforming finite element spaces with a semi-implicit backward Euler discretization. Its magnetic space and discrete energy structure support compactness, stability, and convergence analysis.

3.1. Triangulation and polynomial spaces

  • The triangulation is a regular, quasi-uniform tetrahedral mesh, and the polynomial spaces are defined by total degree on each element.The section introduces these spaces as the basis for the finite element construction.Source: §3.1 · p. 7

3.2. Discrete velocity, pressure, and temperature spaces

  • The velocity, pressure, and temperature spaces use a conforming velocity-pressure pair satisfying a discrete inf-sup condition, with velocity sought directly in the discretely divergence-free space.Pointwise divergence-freeness is assumed only for error analysis, not for well-posedness, stability, or convergence.Source: §3.2 · p. 7

3.3. Discrete space for the magnetic induction

  • The magnetic induction uses H(curl)-conforming Nédélec edge elements, suited to settings where the magnetic field may not belong to H1(D).The magnetic trial space imposes no tangential boundary condition.Source: §3.3 · p. 8

3.4. The Hodge mapping and discrete compactness

  • The discrete solenoidal magnetic space is linked to the continuous one by a Hodge mapping that preserves curl and yields strong L2 compactness for bounded sequences.The mapping also provides spatial integrability required to control the Lorentz force.Source: §3.4 · p. 8

3.5. Discrete inf-sup condition and inverse estimates

  • The discrete inf-sup condition holds with a constant independent of h, and quasi-uniform meshes provide inverse estimates for the finite element spaces.Analogous inverse estimates are used for magnetic and temperature functions when required.Source: §3.5 · p. 9

3.6. The fully discrete scheme

  • The scheme evaluates transport velocities and one magnetic factor at the previous time level while treating dissipative terms, remaining coupled variables, and Coriolis implicitly.Discrete magnetic solenoidality is built into the trial space, so no additional Lagrange multiplier is required.Source: §3.6 · p. 10
  • The semi-implicit backward Euler scheme retains exact electromagnetic cancellation and Coriolis energy neutrality, yielding energy stability rather than energy preservation.The buoyancy terms occur at different time levels and do not cancel.Source: §3 · p. 7, §3.6 · p. 10

4. Well-posedness and stability

The fully discrete scheme is well posed and energy stable, with its proof using the triangular structure and cancellations in the coupled equations. The resulting bounds are independent of the mesh and support the compactness arguments used later.

4.1. Existence of a discrete solution

  • For every admissible mesh and time step, each time level has at least one discrete solution, without a coupling condition between Δt and h.Existence follows by treating the velocity–magnetic subsystem first and then solving the temperature equation.Source: §4.1 · pp. 10–11, §4 · p. 10

4.2. Uniqueness of the discrete solution

  • For prescribed previous-step data, the scheme has at most one solution at each time level, so the fully discrete scheme determines a unique sequence.Subtraction and testing cancel the convective and electromagnetic terms, while strong monotonicity controls the nonlinear diffusion.Source: §4.2 · p. 12

4.3. Energy stability

  • When Δt≤Δt0, the energy estimate has a constant independent of h and Δt, and the required time-step bound is independent of h.The stability argument uses skew-symmetry and electromagnetic cancellation; no CFL-type mesh-time coupling is needed.Source: §4 · p. 10, §4.3 · p. 12

4.4. Consequences of the stability estimates

  • The p-energy estimate also gives an L2-in-space bound for the velocity gradient, supplying compactness estimates for the convergence analysis.The energy bounds, time-interpolant difference control, and gradient estimate support weak compactness and the time-translate argument.Source: §4.4 · p. 13

5. Convergence of the fully discrete scheme

This section establishes convergence of the fully discrete scheme as the mesh width and time step vanish, passing the induction, temperature, and momentum equations to a limit that attains the initial data. A discrete energy limsup inequality and Minty’s argument identify the nonlinear diffusion, subject to an additional condition whose three-dimensional threshold is driven by the Lorentz force.

5.5. Proof of Theorem 5.1

  • Minty’s method identifies the weak flux limit as ν|∇u|^(p−2)∇u using a discrete energy limsup inequality.The condition on p is needed for this identification; compactness and passage to the limit in the other terms hold for every p ≥ 2.Source: §5 · p. 14, §5.2 · p. 14, §5.5 · p. 22

5.3. A time-translate estimate for the velocity

  • Because the p-Laplacian gives no uniform discrete time-derivative bound for velocity, time translates and Simon’s criterion provide the needed compactness.The resulting time-translate estimate is uniform in mesh width and time-step size.Source: §5 · p. 14, §5.3 · p. 15

5.2. Statement of the main result

  • For p ≥ 2 and convergent data, a subsequence converges to a limit satisfying all three equations with a weak limit of the p-Laplacian flux; under (5.1), it is a weak solution.The limit fields belong to the stated energy spaces for velocity, magnetic induction, and temperature.Source: §5.2 · p. 14

These are compatible whenever

  • The domain-independent condition p ≥ 8/3 suffices to identify the flux; p = 2 is also covered, while 2 < p < 8/3 requires additional tools.The stronger restriction arises from the Lorentz force, not convection, and the classical p ≥ 11/5 threshold is insufficient here.Source: §5.2 · p. 14, These are compatible whenever · p. 14

5.4. Time-derivative estimates for the magnetic field and the temperature

  • Dual-norm estimates on the discrete time derivatives of magnetic induction and temperature enable compactness arguments for both fields.These estimates are uniform in mesh width and time-step size.Source: §5.4 · pp. 17–18

5.5. Proof of Theorem 5.1

  • The proof obtains strong L2 space-time convergence of velocity, magnetic induction, and temperature, while recovering the divergence-free constraints for the velocity and magnetic limit.For magnetic induction, compactness is established through the Hodge mapping; temperature converges strongly in L2(0,T;Lq(D)) for 1 ≤ q < 6.Source: §5.5 · pp. 19–20
  • Under (5.1), the Lorentz force is the binding term for placing the time derivative in Lp′(0,T;X′0), allowing the energy equality to be used.The Lions–Magenes lemma gives u ∈ C([0,T];H), making u an admissible test function.Source: §5.5 · p. 23
  • The discrete energy limsup inequality enables Minty’s monotonicity argument to identify the weak limit of the nonlinear diffusion as A(∇u).For p ≥ 2, A is monotone, hemicontinuous, and bounded from Lp(0,T;Lp) into its dual.Source: §5.5 · p. 23
  • The limit (u,b,θ) satisfies the coupled system (2.18)–(2.20), with initial data attained through the time integration-by-parts terms.The equations are first established for the stated test functions and then extended by density.Source: §5.5 · p. 26

6. Error analysis

The error analysis establishes a priori estimates for velocity, magnetic induction, and temperature, with an admissible time step independent of the mesh size. It combines projection-based error equations with consistency estimates designed to preserve the stated spatial and temporal rates.

6.1. Regularity assumption and statement of the main result

  • The estimates are mesh-independent: the admissible time step depends only on the data and T, not on mesh size h.The theorem also gives a unique discrete solution when Δt is below a positive threshold independent of h.Source: §6 · p. 26, §6.1 · p. 26

6.4. Proof of Theorem 6.1

  • Integral representations estimate temporal consistency errors without generating the unwanted h^(2l)/Δt term.The time integral is used instead of endpoint values for the projection consistency error.Source: §6.1 · p. 26

6.1. Regularity assumption and statement of the main result

  • The velocity gradient error is bounded by C*{(Δt)^2 + h^(lp′)}^(1/p), while for p = 2 and l = 1 the spatial rate is first order.The error bound's constant C* is independent of Δt and h.Source: §6.1 · p. 26

6.2. Projections and their approximation properties

  • A linear Stokes projection is used instead of a nonlinear projection, leaving the p-Laplacian nonlinearity in a consistency term.The linear projection is chosen because an L2 estimate is available for it; the nonlinear alternative is treated naturally in quasi-norms.Source: §6.2 · p. 27

6.3. The error equations

  • The error equations compare exact and discrete solutions at t_n, retaining induction and temperature couplings in the magnetic and thermal residuals.The magnetic residual includes the lagged discrete velocity and induction, while the temperature residual includes lagged velocity and temperature terms.Source: §6.3 · p. 27

6.4. Proof of Theorem 6.1

  • The p-Laplacian consistency estimate uses (p, p′)-Young once, producing h^(lp/(p−1)) and the theorem's spatial rate.The estimate is absorbed by the velocity dissipation term.Source: §6.4 · p. 31

7. Numerical experiments

The experiments illustrate the discrete dynamics and energy behavior, then examine temporal and space-time convergence. The observed rates are consistent with the analysis, with spatial rates exceeding the guaranteed exponent for the smooth manufactured solutions.

7.1. Common setting

  • The common setting uses continuous P2 velocity and temperature elements, continuous P1 pressure elements, and second-order Nédélec edge elements for magnetic induction.The domain is the unit cube with a uniform tetrahedral mesh; T = 1 and ν = σ = κ = 1.Source: §7.1 · p. 33

7.2. Qualitative behaviour of the discrete solution

  • At z = 0.25, the velocity forms counter-rotating cells, the magnetic field follows a rotating phase, and the temperature evolves from a hot spot to a cellular pattern and then a cold core.The fields are tracked over t ∈ [0,1] for p = 3.Source: §7.2 · pp. 33–34

7.3. Discrete energy and stability

  • For p = 3 on the fixed mesh, the displayed discrete norms remain bounded and decay on [0,1], consistent with mesh-independent stability.The experiment uses h = 0.433 and examines the time evolution of the discrete norms.Source: §7.3 · p. 34

7.4. Temporal convergence

  • At fixed h = 0.0866 for p = 3, final-time errors converge with slopes close to one, consistent with first-order temporal accuracy.W1,3 is the natural velocity norm; L5 and W1,5 are included as additional diagnostics.Source: §7.4 · p. 34, Figure 5 · p. 38

7.5. Space-time convergence

  • Under coupled refinement ∆t ∼ h^2, observed spatial rates exceed the guaranteed exponent; these rates are measured in h, not temporal accuracy.The authors attribute the higher rates to the smooth manufactured solutions and note that the theoretical bound need not be sharp in this regime.Source: §7.5 · pp. 38–39

8. Conclusion and future directions

The paper analyzes a fully discrete mixed finite element method for rotating incompressible MHD-Boussinesq flows with p-Laplace viscosity. It establishes well-posedness, mesh-independent energy stability, convergence, and error estimates, while identifying open limits on flux identification and the error analysis.

Main conclusions

  • The scheme is well posed for arbitrary h > 0 and ∆t > 0, with a mesh-independent energy estimate and exact cancellation of Lorentz and induction coupling terms.The method uses semi-implicit backward Euler, conforming velocity and temperature elements, and H(curl)-conforming Nédélec elements for magnetic induction.Source: §8 · p. 39

Convergence analysis

  • As h, ∆t → 0, discrete solutions converge along a subsequence, with time-translate compactness and Minty’s method identifying the nonlinear diffusion flux.The flux identification uses a discrete energy limsup inequality.Source: §8 · p. 40

Error estimates

  • The estimate in L∞(0,T;L2) reduces, for p = 2, to the classical rate O(∆t +hl).Source: §8 · p. 40

Future directions

  • Flux identification outside range (5.1) remains open and appears to require additional tools; the spatial error exponent may also be non-sharp for smooth solutions.A more refined treatment of the divergence remainder may relax the pointwise divergence-free condition used for the quadratic Grönwall argument.Source: §8 · p. 40

Declarations

The declarations acknowledge research discussions, identify funding, state that the authors have no relevant competing interests, and describe data availability.

Acknowledgements

  • The authors acknowledge Dr. Kim Ngan Le for discussions, comments, and suggestions during the work’s development.Source: Declarations · p. 40

Funding

  • The work was supported by the Department of Science and Technology, Government of India, through the INSPIRE Faculty Award, grant IFA21-MA 158.Source: Declarations · p. 40

Conflict of interest

  • The authors declare no competing interests relevant to the content of the article.Source: Declarations · p. 40

Data availability

  • The numerical data were generated through the Section 7 finite element simulations; additional computational data are available from the corresponding author upon reasonable request.Source: Declarations · p. 40
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