Finite Element Approximation of the Enthalpy Formulation for Stefan Problems on Evolving Surfaces

Jul 16, 2026

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A spatially discrete evolving surface finite element method is developed for approximating the enthalpy formulation of a two-phase Stefan problem posed on an evolving surface. The method avoids mass-lumping and discrete maximum principles, proving numerical stability. It establishes O(√h) L2 error bounds for temperature under minimal regularity via a new projection-type operator. Implementation details include an exact discretisation to prevent numerical quadrature errors, along with numerical experiments and convergence demonstrations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages32

Reading Time1 h 21 min

Introduction

Problem setting and enthalpy formulation

Motivation and related work

Main contributions

arXiv :2607 .03219v1 [math .NA] 3 Jul 2026 FINITE ELEMENT APPROXIMATION OF THE ENTHALPY FORMULATION FOR STEFAN PROBLEMS ON EVOLVING SURFACES PHILIP J. HERBERT, THOMAS SALES, AND CHANDRASEKHAR VENKATARAMAN Abstract. We propose, and analyse, a spatially discrete evolving surface finite element method for the approximation of the enthalpy formulation of the two-phase Stefan problem posed on an evolving surface. Our approach does not rely on mass-lumping and discrete maximum principles. We prove this numerical method is numerically stable, and prove O ( √h) error bounds for the temperature in the L2L2 norm under minimal regularity assumptions by introducing a new projection-type operator. We complement our analysis with discussion on the implementation of this numerical method, where we propose a novel implementation that avoids errors due to numerical quadrature and which has not previously been considered in the literature even in the stationary, flat setting. We also include numerical experiments and experimental order of convergence demonstrations. 1. Introduction We are interested in the analysis of evolving surface finite element methods (ESFEM) for twophase Stefan problems posed on an evolving surface, Ω(t) ⊂ R3 , moving with a prescribed velocity, V. In particular, we consider the so-called enthalpy formulation of the Stefan problem, (1.1a) (1.1b) ∂•e + e(∇Ω · V) − ∆Ω u = f, on Ω(t), e ∈ β (u), with initial condition e(0) = e0 ∈ L2 (Ω(0)) and where the enthalpy β is a set-valued map, as discussed in Section 2. Here the function e is the enthalpy function (i.e. the heat content) and u is the temperature distribution. We shall assume that Ω(t) is a sufficiently smooth surface without boundary, and hence there are no boundary conditions associated with (1.1) . We defer discussion of the differential operators used in (1.1) until Section 2. Problems of the form (1.1) were proposed and analysed in [4], wherein the authors study the well-posedness for suitably defined solutions. Stefan-type problems posed on a surface appear in nature as models of phase transitions on surfaces, for example when a soap bubble freezes [1, 34], in free boundary limits of bulk-surface models of ligand-receptor dynamics [3, 28], and in models of cell-polarisation [39, 46] . There is also interest in industrial applications of Stefan problems on surfaces, such as welding [15, 44 , 45] and aircraft icing [35, 54], which can be modelled as Stefan problems on surfaces (which may also be deforming) . This enthalpy formulation provides a generalised notion of solution for the Stefan problem and has been used extensively since initial work in the 1960s by Ole˘ınik [53], Kamenomostskaja [37], and Friedman [33] . This weak notion of solution allows one to implicitly capture the evolution of the free boundary, Γ(t) as illustrated in Figure 2 , through the definition of the graph β (sometimes referred to as the generalised enthalpy [25]), instead of explicitly capturing the free boundary, as in front-tracking methods [13] . We note that 2020 Mathematics Subject Classification. 65M60, 65M15, 35R35, 80A22 . Key words and phrases. Stefan problem, free boundary problem, evolving surface finite elements, enthalpy formulation, degenerate PDE. 1 2 P. J. HERBERT, T. SALES, AND C. VENKATARAMAN sufficiently regular enthalpy solutions do indeed solve the more standard strong formulation of the Stefan problem (2.7), provided the free boundary, Γ(t), does not develop an interior (a so-called mushy region), cf. [4, Remark 2.12] . We refer the reader to [16, 25 , 36 , 58 , 59] for further details on the enthalpy formulation of the Stefan problem, as well as [19, 24 , 47 , 48 , 50 , 51 , 52] for resultson finite element approximations of the two-phase Stefan problem. To our knowledge, there is no literature concerning the numerical approximation of Stefan-type problems posed on an evolving surface. We do however, refer the reader to recent work by

FAQs

What problem does the paper study and where is it posed?

The paper studies a two-phase Stefan problem in an enthalpy formulation posed on an evolving surface Ω(t) ⊂ R3 moving with prescribed velocity.

What are the key advantages of the proposed evolving surface finite element method?

The approach avoids mass-lumping and discrete maximum principles, proves numerical stability, and derives O(√h) error bounds for temperature in the L2 norm.

How does the paper handle implementation issues such as numerical quadrature errors?

It proposes an exact discretisation for the fully discrete problem (under piecewise polynomial β) that prevents errors due to numerical quadrature, and it supports the claims with numerical experiments and convergence-rate demonstrations.

Jul 16, 2026

Upload by Eliana