We describe some differential geometric bifurcation problems and their treatment in the Matlab continuation and bifurcation toolbox pde2path. The continuation steps consist in solving the PDEs for the normal displacement of an immersed surface $X ⊂\mathbb{R}^3,$ with bifurcation detection and possible subsequent branch switching. The examples include minimal surfaces such as Enneper’s surface and a Schwarz-P-family, some non-zero constant mean curvature surfaces such as liquid bridges, and some 4th order biomembrane models. In all of these we find interesting symmetry-breaking bifurcations. A few of these are (semi)analytically known and hence used as benchmarks.

An extended Courant element is constructed on an $n$ dimensional polytope $K,$ which reduces to the usual Courant element when $K$ is a simplex. The set of the degrees of freedom consists of function values at all vertices of $K,$ while the shape function space $P_K$ is formed by repeatedly using the harmonic extension from lower dimensional face to higher dimensional face. Several fundamental estimates are derived on this element under reasonable geometric assumptions. Moreover, the weak maximum principle holds for any function in $P_K,$ which enables us to use the element for approximating an obstacle problem in three dimensions. The corresponding optimal error estimate in $H^1$-norm is also established. Numerical results are reported to verify theoretical findings.

Driven by the challenging task of pursuing the robust and accurate iterative numerical solution of the three-dimensional flux-limited multi-group radiation diffusion equations in an efficient and scalable manner, we propose and analyze a generalized matrix splitting preconditioning scheme with two selective relaxations and algebraic multigrid subsolves, introduce an algebraic quasi-optimal choice strategy to determine the involved parameters and consider its sequential implementation and two-level parallelization. A great deal of numerical results for typical unstructured twenty-group problems arising from realistic simulations of the hydrodynamic instability are presented and discussed to demonstrate the robustness, efficiency, strong and weak parallel scaling properties with up to 2,816 parallel processor cores together with the competitiveness of the proposed preconditioner when compared with several state-of-the-art monolithic and block preconditioning approaches.

For the primal-dual monotone inclusion problem, the split-Douglas-Rachford (SDR) algorithm is a well-known first-order splitting method. Similar to other first-order algorithms, the efficiency of SDR is greatly influenced by its step parameters. Therefore, expanding the range of stepsizes may lead to improved numerical performance. In this paper, we prove that the stepsize range of SDR can be expanded based on a series of properties of the product Hilbert space. Moreover, we present a concise counterexample to illustrate that the newly proposed stepsize range cannot be further enhanced. Furthermore, to bridge the theoretical gap in the convergence rate of SDR, we analyze the descent of SDR’s fixed point residuals and provide the first proof of a sublinear convergence rate for the fixed point residuals. As an application, we utilize SDR to solve separable convex optimization problems with linear equality constraints and develop a novel preconditioned alternating direction method of multipliers (NP-ADMM). This method can handle scenarios where two linear operators are not identity operators. We propose strict convergence conditions and convergence rates for the preconditioned primal-dual split (P-PDS) method and NP-ADMM by demonstrating the relationship between SDR, P-PDS, and NP-ADMM. Finally, we conduct four numerical experiments to verify the computational efficiency of these algorithms and demonstrate that the performance of these algorithms has been significantly improved with the improved conditions.

In this paper, we investigate the numerical solution of the two-dimensional fractional Laplacian wave equations. After splitting out the Riesz fractional derivatives from the fractional Laplacian, we treat the Riesz fractional derivatives with an implicit scheme while solving the rest part explicitly. Thanks to the tensor structure of the Riesz fractional derivatives, a splitting alternative direction implicit (S-ADI) scheme is proposed by incorporating an ADI remainder. Then the Gohberg-Semencul formula, combined with fast Fourier transform, is proposed to solve the derived Toeplitz linear systems at each time integration. Theoretically, we demonstrate that the S-ADI scheme is unconditionally stable and possesses second-order accuracy. Finally, numerical experiments are performed to demonstrate the accuracy and efficiency of the S-ADI scheme.

Recently, the tensor nuclear norm, based on self-supervised nonlinear transformations, has gained significant attention in multidimensional image restoration. However, its primary concept involves solely nonlinear transformations along the third mode of a three-order tensor, which limits its flexibility in dealing with correlations in various modes of high-dimensional data. This paper makes three main contributions. Firstly, we introduce a novel approach called three-directional self-supervised nonlinear transform tensor nuclear norm (3DSTNN), which takes into account nonlinear transformations in all modes and can better represent the global structure of the tensor. Secondly, we suggest a model for multidimensional picture recovery that minimizes ranks by modeling the underlying tensor data as low-rank components subjected to nonlinear transformations. Thirdly, to solve the suggested model, we create an effective algorithm based on the alternating direction method of multipliers (ADMM). In low-rank tensor approximation for image restoration, our approach performs better than the state-of-the-art, according to extensive experimental results on both synthetic and actual datasets.

Fourier stability analysis works well and is popular for the finite difference schemes of the linear partial differential equations. However, there are less works on the Fourier convergence analysis, and many of the existing ones require unreasonable assumptions. After removing the assumptions, we provide rigorous Fourier convergence analyses for the equation with one time fractional derivative in our previous work. In the current work, by using different ideas, we propose the rigorous Fourier convergence analyses for the equation with several time fractional derivatives, i.e., the Fokker-Planck equation of tempered fractional Langevin-Brownian motion, still without the strong assumptions. The numerical experiments are performed to confirm the theoretical results.

A linear and second order numerical scheme is proposed and analyzed for a droplet thin film coarsening model, with a singular Leonard-Jones energy potential involved. This numerical scheme with unconditional energy stability is based on the backward differentiation formular (BDF) method in time derivation combining with the stabilized scalar auxiliary variable approach in time and the Fourier spectral method in space. A second order accurate artificial regularization term, in the form of $S∆(\phi^{n+1}− 2\phi^n + \phi^{n−1}),$ is added in the numerical scheme to make better stability of the numerical scheme. Moreover, we present the detail proof for unconditional energy stability property of the numerical scheme, without any restriction for the time step size. In addition, an $\mathcal{O}(∆t^2+h^m)$ rate convergence estimate in the $ℓ^∞(0, T ; ℓ^2)$ norm are derived in details with the help of a priori assumption for the error at the previous time step. Some numerical simulation results are presented to demonstrate the efficiency and accuracy of the second-order numerical scheme.

It is known that the solution of the Cauchy problem for partial differential equations of hyperbolic type can be reduced to singular integrals of a unique form. Laterly, singular integrals were called integrals in the sense of Hadamard or Hadamard integrals. In addition to equations of the hyperbolic type, Hadamard integrals are widely used in elasticity, electrodynamics, aerodynamics, and other vital areas of mechanics and mathematical physics. The exact evaluation of Hadamard integrals is possible only in exceptional cases, so that there is a need to develop approximate methods for their evaluation. In the present paper, we develop an optimal algorithm for the approximate calculation of Hadamard integrals. Here, we deal with finding the analytical form of the coefficients of an optimal quadrature formula. Numerical results show the validity and accuracy of the method.

In this paper, a priori error estimate of the nonconforming virtual element method with Streamline Upwind/Petrov-Galerkin (SUPG) stabilization for convection dominated diffusion equation is discussed. The discrete scheme of the nonconforming virtual element method is constructed and a priori error estimate is derived with a norm containing jump term. Numerical experiments verify the a priori error estimate in a set of polygonal meshes.