Convergence analysis of a new dynamic diffusion method

Nonlinear diffusion methods have proved to be powerful methods in the processing of 2D and 3D images. They allow a denoising and smoothing of image intensities while retaining and enhancing edges. On the other hand, compression is an important topic in image processing as well. Here a method is presented which combines the two aspects in an efficient way. It is based on a semi-implicit Finite Element implementation of nonlinear diffusion. Error indicators guide a successive coarsening process. This leads to locally coarse grids in areas of resulting smooth image intensity, while enhanced edges are still resolved on fine grid levels. Special emphasis has been put on algorithmical aspects such as storage requirements and efficiency. Furthermore, a new nonlinear anisotropic diffusion method for vector field visualization is presented.

Virtual element method for semilinear elliptic problems on polygonal meshes

Conforming virtual element approximations of the two-dimensional Stokes problem

Neumann problems of 2D Laplace's equation by method of fundamental solutions

Adaptive h- p procedures for high accuracy finite element analysis of two-dimensional linear elastic problems

In this paper, we apply semi-Lagrangian discontinuous Galerkin (SLDG) methods for linear hyperbolic equations in one space dimension and analyze the error between the numerical and exact solutions under the L 2 L^2 -norm. In all the previous works, the theoretical analysis of the SLDG method would suggest a suboptimal convergence rate due to the error accumulation over time steps. However, numerical experiments demonstrate an optimal convergence rate and, if the terminal time is large, a superconvergence rate. In this paper, we will prove optimal convergence and optimal superconvergence rates. There are three main difficulties: 1. The error analysis on overlapping meshes. Due to the nature of the semi-Lagrangian time discretization, we need to introduce the background Eulerian mesh and the shifted mesh. The two meshes are staggered, and it is not easy to construct local projections and to handle the error accumulation during time evolution. 2. The superconvergence of time-dependent terms under the L 2 L^2 -norm. The error of the numerical and exact solutions can be divided into two parts, the projection error and the time-dependent superconvergence term. The projection strongly depends on the superconvergence rates. Therefore, we need to construct a sequence of projections and gradually improve the superconvergence rates. 3. The stopping criterion of the sequence of projections. The sequence of projections are basically of the same form. We need to show that the projections exist up to some certain order since the superconvergence rate cannot be infinity. Hence, we will seek some “hidden” condition for the existence of the projections. In this paper, we will solve all the three difficulties and construct several local projections to prove the optimal convergence and superconvergence rates. Numerical experiments verify the theoretical findings.

A numerical method for a two-dimensional curl–curl and grad-div problem is studied in this paper. It is based on a discretization using weakly continuous P 1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive $${\epsilon}$$) in both the energy norm and the L 2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

In this paper, Newmark time-stepping scheme and edge elements are used to numerically solve the time-dependent scattering problem in a three-dimensional cavity. Finite element methods based on the variational formulation derived in [23] are considered. Due to the lack of regularity ofer, the existence and uniqueness of the discrete solutions and their convergence are proved by using the concept of collectively compact operators. An optimal convergence rate in the energy norm is also established.

A third‐order singularly perturbed problem of reaction–diffusion type is solved numerically by the local discontinuous Galerkin (LDG) method. By using the local Gauss–Radau projection for the primal variable u, but local L2 projection for the two auxiliary variables p and q, we obtain an optimal convergence rate of order (or up to a logarithmic factor) in the energy norm, uniformly in the singular perturbation parameter. Here is the degree of the piecewise polynomials used in the discrete space and N is the number of mesh intervals. The layer‐adapted meshes are used including a standard Shishkin mesh, a Bakhvalov–Shishkin mesh, and a Bakhvalov‐type mesh. Numerical experiments are given to show the sharpness of our theoretical results.

In this paper, we use the mixed direct discontinuous Galerkin method (DDG) to solve the biharmonic equation. Firstly, by introducing an auxiliary variable, the biharmonic equation is split into two second-order equations. Secondly, the variational problem based on the DDG method of the system is derived and its well-posedness is proven. Next, error estimates of the approximate solution in L 2 norm and energy norm are present. For a given polynomial degree k (k ≥ 1), the optimal convergence rates concerning energy norm and norm are k and k + 1, respectively. Finally, numerical results demonstrate the accuracy and capability of the method.

Locally discontinuous but globally continuous Galerkin methods for elliptic problems

The main focus of this paper is to improve the performance of the texture model Optimized Local Ternary Patterns (OLTP), which is known as one of the successful variants of the texture model Local Binary Patterns (LBP), a well known method for texture analysis and its applications. Generally preprocessing is used in digital image processing for reducing the unwanted noise and disturbances in such a way that it improves the quality of the image. Preprocessing not only removes the distortions but also enhances the features of the image for further processing. To achieve better recognition accuracy, the texture model OLTP was combined with a preprocessing method that uses nonlinear diffusion method as a preprocessing tool in this paper, with the hope that this idea will surely improve the local feature description and texture classification process. This nonlinear diffusion method uses two newly developed edge stopping functions for preprocessing. This proposed method is tested with two standard texture datasets namely Brodatz Dataset and Usptex dataset. The results show that the use of the preprocessing step really improved the texture classification accuracy.

In this paper, we propose a stabilizer free spatial weak Galerkin (SFSWG) finite element method for solving time-dependent convection diffusion equations based on weak form Eq. (4). SFSWG method in spatial direction and Euler difference operator Eq. (37) in temporal direction are used. The main reason for using the SFSWG method is because of its simple formulation that makes this algorithm more efficient and its implementation easier. The optimal rates of convergence of 𝒪⁢(hk) and 𝒪⁢(hk+1) in a discrete H1 and L2 norms, respectively, are obtained under certain conditions if polynomial spaces (Pk⁢(K),Pk⁢(e),[Pj⁢(K)]2) are used in the SFSWG finite element method. Numerical experiments are performed to verify the effectiveness and accuracy of the SFSWG method.

We propose and analyze an unfitted finite element method of arbitrary order for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. We apply a non-symmetric bilinear form and the boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. The optimal convergence rate under the energy norm is derived, and \(O(h^{-2})\)-upper bounds of the condition numbers are shown for the final linear systems. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.

A discontinuous Galerkin least-squares method for div–curl systems