Superconvergence of the function value for pentahedral finite elements for an elliptic equation with varying coefficients

In this article, for an elliptic equation with varying coefficients, we first derive an interpolation fundamental estimate for the $\mathcal{P}_{2}(x,y)\otimes \mathcal{P}_{2}(z)$P2(x,y)⊗P2(z) pentahedral finite element over uniform partitions of the domain. Then combined with the estimate for the $W^{2,1}$W2,1-seminorm of the discrete Green function, superconvergence of the function value between the finite element approximation and the corresponding interpolant to the true solution is given.

On the Taut String Interpretation and Other Properties of the Rudin–Osher–Fatemi Model in One Dimension

We study the one-dimensional version of the Rudin–Osher–Fatemi (ROF) denoising model and some related TV-minimization problems. A new proof of the equivalence between the ROF model and the so-called taut string algorithm is presented, and a fundamental estimate on the denoised signal in terms of the corrupted signal is derived. Based on duality and the projection theorem in Hilbert space, the proof of the taut string interpretation is strictly elementary with the existence and uniqueness of solutions (in the continuous setting) to both models following as by-products. The standard convergence properties of the denoised signal, as the regularizing parameter tends to zero, are recalled and efficient proofs provided. The taut string interpretation plays an essential role in the proof of the fundamental estimate. This estimate implies, among other things, the strong convergence (in the space of functions of bounded variation) of the denoised signal to the corrupted signal as the regularization parameter vanishes. It can also be used to prove semi-group properties of the denoising model. Finally, it is indicated how the methods developed can be applied to related problems such as the fused lasso model, isotonic regression and signal restoration with higher-order total variation regularization.