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Finite Element Approximation of a Hemivariational Inequality for Steady-State Heat Conduction: Double-Limit Convergence of Penalization and Discretization

Piotr Bartman-Szwarc, Anna Ochal, Domingo A. Tarzia

arXiv:2609.35538v1math.NAcs.NAmath.AP

TL;DR

For heat conduction with mixed boundary conditions, the paper addresses the finite element approximation of a hemivariational inequality with a penalized Dirichlet boundary condition. It combines mesh refinement and increasing penalization, proving strong convergence to the limit solution; uniform-in-α error estimates for the penalized problems are not expected.

  • Problem

    The simultaneous finite element and penalty limit for a hemivariational inequality with a penalized Dirichlet condition on part of the boundary had not been addressed in the literature.

    Source: §1 · p. 3

  • Method

    The paper combines finite element discretization with penalization of the Dirichlet boundary condition and analyzes h→0 and α→∞ together.

    Source: §1 · p. 3

  • Results

    Discrete solutions converge strongly to the unique limit solution as (h, α) tends to (0, ∞).

    Source: §1 · p. 3

  • Takeaways & Limitations

    The double-limit result extends beyond earlier results confined to a quadratic potential to allow nonconvex and nonsmooth boundary laws.

    Source: §1 · p. 3

    No error estimate uniform in α is expected for the penalized problems because the existing theory's smallness condition fails as α grows.

    Source: §3 · p. 15

Abstract

from arXiv · show

In this paper we study the numerical approximation and convergence analysis of a steady-state heat conduction problem with mixed boundary conditions. The physical model is governed by a hemivariational inequality depending on a heat transfer parameter $\alpha > 0$. We consider the finite element approximation of this penalized problem, as well as its corresponding limit problem with a prescribed constant temperature on a part of the boundary. The main theoretical result establishes the strong convergence of the discrete solutions. Specifically, we prove the double-limit convergence of the finite element approximations to the limit solution as the mesh size $h$ tends to zero and the penalization parameter $\alpha$ tends to infinity, independently and simultaneously. We further complement the convergence analysis with an error estimate of optimal order for the discrete limit problem, and we explain why an estimate uniform in the penalization parameter cannot be expected. The theoretical results are illustrated by numerical simulations on four examples verified by the method of manufactured solutions.

1 Introduction

The paper studies finite element approximation of heat conduction with nonsmooth, potentially nonmonotone boundary laws, combining mesh refinement with a penalty limit that recovers a prescribed boundary temperature. Its main contribution is strong convergence in this simultaneous limit, alongside numerical validation and an explanation of why penalized-problem error estimates cannot be uniform in the penalty parameter.

1 Introduction

  • The boundary law on Γ3 is a nonlinear Robin condition governed by the Clarke subdifferential, while the limit problem prescribes temperature b there.Hemivariational inequalities provide a framework for nonsmooth boundary laws, including nonlinear, nonmonotone, or multivalued heat exchange.Source: §1 · p. 2
  • The penalty replaces the hard temperature constraint on Γ3 with a Robin-type condition whose flux is α∂j(u), recovering the Dirichlet condition as α grows.Earlier work addressed the Robin-to-Dirichlet limit for nonlinear hemivariational problems without the finite element double limit studied here.Source: §1 · p. 2
  • The simultaneous limit cannot be handled by freezing either parameter, and the growing penalty term requires compactness rather than monotonicity.The penalty term is upper semicontinuous, while the discrete solutions and test functions vary together.Source: §1 · p. 3
  • No error estimate uniform in α is expected for the penalized problems, while the discrete limit problem admits an optimal-order estimate.The smallness condition behind existing theory degenerates as α tends to infinity, and finite-α discrete problems may have multiple solutions.Source: §1 · p. 3
  • Discrete solutions converge strongly to the unique limit solution as h tends to zero and α tends to infinity simultaneously.The paper identifies this double-limit result as its main contribution.Source: §1 · pp. 2–3
  • The paper reports numerical experiments that validate the theoretical findings and compute convergence rates explicitly.No numerical rate values are given in these passages.Source: §1 · pp. 3–4

2 Preliminaries

The preliminaries set up the function spaces, assumptions, and weak formulations for the penalized and limit problems. They also establish a convergence property for the hemivariational term that is used to pass to limits in the later analysis.

2 Preliminaries

  • The spaces K and K0 encode the prescribed boundary values, while the bilinear form a is continuous and coercive on V0.K imposes temperature b on Γ3; K0 imposes zero there, and both require zero trace on Γ1.Source: §2 · p. 4
  • The limit problem is a variational equality on K, whereas the penalized problem takes the form of a hemivariational inequality.The weak formulation expresses the limit problem using tests in K0 or, equivalently, comparisons within K.Source: §2 · p. 5
  • The hypotheses bound the Clarke subdifferential's growth and constrain the directional derivative toward b, with equality only at r = b.These sign and equality conditions help identify the boundary behavior of limits of penalized solutions.Source: §2 · p. 5
  • Under the stated data and superpotential assumptions, each penalized problem has a solution, and the limit problem has a unique solution.The penalized existence result requires H(j)(a)–(c); the stated limit-problem uniqueness result assumes H0.Source: §2 · p. 5
  • A convergence property for the hemivariational term supports passage to the limit when h tends to zero, α tends to infinity, or both do so together.Its proof uses compactness of the trace and upper semicontinuity of the generalized directional derivative.Source: §2 · p. 6

3 Discretization by Finite Element Method

This section presents the conforming P1 finite element approximation and analyzes strong convergence under mesh refinement, increasing penalization, and their simultaneous limit, along with optimal-order convergence for the discrete limit problem. It also notes that the stated basic assumptions alone do not guarantee a convergence rate for the finite element approximation and describes a compactness-based approach to the double-limit proof.

3 Discretization by Finite Element Method

  • The method uses degree-one Lagrange elements on boundary-resolving triangulations, making the approximation conforming; the discrete convex sets approximate K in the sense of Mosco.The Mosco property comprises strong approximation of each v∈K and weak-limit closure in K.Source: §3 · pp. 6–7
  • The convergence analysis completes a diagram connecting the continuous and discrete problems through limits in h, α, and (h, α).It focuses on convergence of the discrete limit solutions as h→0 and the three stated regimes for penalized discrete solutions.Source: §3 · p. 8
  • The discrete limit solution is the best approximation to u∞ in Kh in the V0 norm, yielding an optimal-order error estimate for the conforming approximation.The estimate is stated under the regularity conditions used for interpolation in the proof.Source: §3 · pp. 9–10
  • Under the stated solution regularity, discrete solutions converge strongly in all three regimes: fixed α as h→0, fixed h as α→∞, and jointly as (h, α)→(0, ∞).The joint limit is the unique solution of (S∞).Source: §3 · p. 10
  • Under (H0) and H(j) alone, no rate is guaranteed because (Sα) need not have a unique solution.An estimate for an arbitrarily selected solution may be uninformative; Section 4 gives data for which (Sh α) has two strict local minimizers throughout an interval of α.Source: §3 · p. 15
  • The smallness condition α m_j c^2 < m_a, with c the trace-embedding norm, fails for m_j > 0 when α ≥ m_a/(m_j c^2), obstructing the limit α → +∞.Theorem 6 rests on compactness and (1), while Proposition 5 has no counterpart uniform in α.Source: §3 · p. 15

4 Numerical Results

The numerical experiments support convergence when mesh refinement and increasing penalization are coordinated, and illustrate the distinct error sources and finite-parameter behaviors across four examples. They also show optimal-order convergence for the limit problem and non-uniqueness for one discrete example.

Example 1

  • The penalization error is exactly order α^-1, while the P1 discretization error is order h^(r−1); balancing them gives an optimal-order double limit.Neither error vanishes when its corresponding parameter is held fixed.Source: §4 · p. 16
  • Only the α = ∞ route and a coupled route with α^-1 and h decreasing at comparable rates converge to u∞; the other routes level off at their error floors.The coupled route converges with slope 1, with a larger constant than the α = ∞ route.Source: §4 · p. 17, Figure 2 · p. 17

Example 2

  • Example 2’s measured rates are about 0.54 in V and 1.07 in L2, consistent with predicted rates 1/2 and 1, respectively.The final refinement is affected by error in the reference solution.Source: §4 · pp. 17–18

Examples 3 and 4

  • For Examples 3 and 4, only the fraction with overshoot exceeding δ falls with α; at α = 10^3, similar above-b fractions mask distinct traces.Example 4’s trace is pinned to b while Example 3’s remains above it.Source: §4 · pp. 18–19, Figure 3 · p. 20
  • Example 3 has two discrete solutions over a finite α window, while five starting points agree to 10^-14 for Example 4 on either side of its transition.The two Example 3 branches differ substantially in both solution and energy; the multiplicity is attributed to the superpotential and data, not the mesh or optimization procedure.Source: §4 · p. 19, Table 2 · p. 19, Figure 4 · p. 21

Convergence in h

  • For Example 1 and the limit problem of Examples 3 and 4, measured rates are 1.00 in V and 2.00 in L2; Example 2’s V rate is 0.54 against the predicted 1/2.The V rates verify the convergence theory, and the Example 2 exponent r−1 is attained and cannot be improved.Source: §4 · pp. 20–21

Examples 3 and 4

  • Example 3’s two finite-α solutions are strict local minimizers, so the authors explain that a uniform-in-α error estimate is out of reach.The study does not determine whether the pinned branch is a second local minimum or a saddle, and claims no rate from branch-varying comparisons.Source: §4 · pp. 20, 22
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