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Abstract The paper introduces a finite element method for an Eulerian formulation of partial differential equations governing the transport and diffusion of a scalar quantity in a time-dependent domain. The method follows the idea from[C. Lehrenfeld and M. Olshanskii,An Eulerian finite element method for PDEs in time-dependent domains,ESAIM Math. Model. Numer. Anal. 53 2019, 2, 585–614]of a solution extension to realise the Eulerian time-stepping scheme. However, a reformulation of the partial differential equation is suggested to derive a scheme which conserves the quantity under consideration exactly on the discrete level. For the spatial discretisation, the paper considers an unfitted finite element method. Ghost-penalty stabilisation is used to realise the discrete solution extension and gives a scheme robust against arbitrary intersections between the mesh and geometry interface. The stability is analysed for both first- and second-order backward differentiation formula versions of the scheme. Several numerical examples in two and three spatial dimensions are included to illustrate the potential of this method. 

ABSTRACT This work proposes a novel approach for coupling non‐isothermal fluid dynamics with fracture mechanics to capture thermal effects within fluid‐filled fractures accurately. This method addresses critical aspects of calculating fracture width in enhanced geothermal systems, where the temperature effects of fractures are crucial. The proposed algorithm features an iterative coupling between an interface‐capturing phase‐field fracture method and interface‐tracking thermo‐fluid‐structure interaction using arbitrary Lagrangian–Eulerian coordinates. We use a phase‐field approach to represent fractures and reconstruct the geometry to frame a thermo‐fluid‐structure interaction problem, resulting in pressure and temperature fields that drive fracture propagation. We developed a novel phase‐field interface model accounting for thermal effects, enabling the coupling of quantities specific to the fluid‐filled fracture with the phase‐field model through the interface between the fracture and the intact solid domain. We provide several numerical examples to demonstrate the capabilities of the proposed algorithm. In particular, we analyze mesh convergence of our phase‐field interface model, investigate the effects of temperature on crack width and volume in a static regime, and highlight the method's potential for modeling slowly propagating fractures.