We develop and analyze a structure-preserving discretization for the coupled Stokes–Biot equations. The method is based on a fully-mixed formulation and computes directly the fluid and poroelastic stresses and the Darcy velocity. The scheme employs equal-order polynomial approximations for all unknowns, yet exhibits local momentum conservation, strongly symmetric stress fields, and exactly divergence-free velocity in the fluid domain. Transmission conditions enforcing mass conservation, stress balance, and the Beavers–Joseph–Saffman slip with friction law are incorporated directly into the formulation as natural interface conditions through careful balance of finite element spaces across subdomains. Well-posedness of the fully discrete scheme based on the backward Euler time-stepping method is established, followed by optimal convergence error estimates for all variables. The analysis shows that the method is locking-free in the regimes of small storativity and almost incompressible material. Several numerical experiments are carried out to validate the accuracy and robustness of the proposed scheme.
We analyze a staggered finite difference approximation for a hybrid PDE-ODE transmission system consisting of two one-dimensional wave equations coupled through a dynamic interior mass. The continuous model is considered in an interface-damped regime in which the right boundary is conservative and the only dissipation acts through the transmission condition. In this regime, exponential stability is conditional and is governed by an arithmetic nonresonance condition on the geometry and physical coefficients. The main question addressed in this paper is whether this conditional stability mechanism is preserved at the semi-discrete level. We construct a finite difference scheme adapted to the dynamic transmission law and prove that it preserves the discrete energy dissipation identity. We then derive a mesh-dependent spectral nonresonance condition characterizing the absence of purely imaginary eigenvalues of the semi-discrete generator. For each fixed mesh, this condition yields exponential stability of the semi-discrete semigroup. Our main result concerns the decay rate under mesh refinement: it shows that, when the travel-time ratio of the two strings is a rational number satisfying the arithmetic condition and the meshes are matched so that the numerical travel time per cell is the same on both strings, the semi-discrete generators satisfy a mesh-uniform resolvent estimate on the imaginary axis; consequently the semi-discrete energy decays exponentially at a rate independent of the mesh size. The proof combines an exact discrete Rellich identity, explicit transfer identities for the two strings, a factorization of the interface force produced by the half-cell inertia, and a parity argument on the matched family. We further prove that the discrete nonresonance condition is branchwise consistent with the continuous arithmetic condition and that resonant discrete eigenvalues converge to the corresponding continuous eigenvalues. Numerical experiments illustrate the resonant and nonresonant regimes, the mesh-independent separation of the computed spectral abscissa from the imaginary axis on matched meshes predicted by the theory, and its loss on unmatched meshes.
Inverse imaging problems such as denoising and deblurring aim to recover a clean image from corrupted measurements by incorporating prior knowledge. Classical regularizers like Tikhonov and total variation are mathematically convenient, yet they often oversimplify the complex structure of natural images. Regularization by Denoising (RED) uses a learned denoiser and yields strong theoretical guarantees, but typical implementations rely on full-batch updates that become computationally costly for large-scale problems. In this paper, we propose two accelerated stochastic formulations of RED that combine RED’s fidelity with the efficiency of stochastic optimization. In the direct formulation, we develop a stochastic primal–dual fixed‑point algorithm that computes mini‑batch gradients, applies variance reduction and uses a relaxation parameter for inertial updates. In the constrained formulation (RED‑PRO), we introduce a complementary algorithm that employs adaptive dual scaling and projection onto the denoiser’s fixed-point set to reduce denoiser calls. Both formulations dispense with strong convexity in favour of Polyak–Łojasiewicz, quadratic-growth and metric subregularity conditions, enabling linear convergence under realistic assumptions. A unified analysis proves convergence, and extensive experiments show that both stochastic RED formulations match or surpass state-of-the-art methods while significantly reducing computational cost and maintaining high PSNR and SSIM.
This work presents a numerical analysis of the problem studied in [23] with a numerical simulation. More precisely, we develop a first-order semi-implicit finite-difference method with convex-concave splitting for the N-component Cahn–Hilliard equation with constant mobility and singular potentials. The existence of a numerical solution relies on a positivity-preserving method for handling equations with singular potentials, ensuring uniqueness of the numerical solution. We establish that the first-order scheme is unconditionally energetically stable, meaning that the associated discretized energy functional decreases over time. This stability result, along with a uniform $H^1$ bound on the order-parameter, is primarily attributed to the convex-concave splitting. In our numerical approach, convex terms are treated implicitly using Newton–Raphson, while concave terms are handled explicitly. Furthermore, we conduct a 2D ternary (N = 3) simulation with random initial data similar to that appearing in [18]. The simulation demonstrates mass conservation, indicating that the concentration of each component is preserved. Additionally, we investigate Eyre’s ”cascades of spinodal decomposition” [18], showing numerically that ”cascading” is more pronounced in Tsallis entropy density functionals with q-values just under 1.
We study finite element approximations for the dynamics of a linear elastic body with a frictional contact interface. The tangential motion is governed by the Tresca friction law, while the normal contact is described by a modified Signorini condition involving both displacement and velocity through a parameter δ . This condition formally includes the usual nonpenetration condition when δ = 0 and the Signorini condition on velocity in the limit δ = ∞ . We first propose a semi-discrete finite element scheme based on a mass-lumping treatment of the contact terms. The scheme is formulated both as a variational inequality and as an equivalent Lagrange multiplier system. Its well-posedness is proved by a regularization argument, and stability and error estimates are derived. We then introduce a fully discrete scheme and establish its well-posedness, stability, and convergence under a suitable time-step restriction. Finally, we develop several iterative methods, including an Uzawa algorithm, an active/inactive set algorithm, and an augmented Lagrangian algorithm. Numerical experiments are carried out to test the convergence rate of the proposed scheme and the efficiency of the iterative algorithms.
