Mortar Spectral Element Method Research Articles

Numerical analysis on the mortar spectral element methods for Schrödinger eigenvalue problem with an inverse square potential

In this paper, we present an hp analysis of the mortar spectral element method for the Schrödinger eigenvalue problem (−Δ+c2‖x‖2)u=λu, and thereby justify the numerical findings in [30], where the method was demonstrated to be efficient to handle the singularities arising from both the inverse square potential and the reentrant/obtuse corners with exponential order of convergence. Non-uniformly weighted Sobolev spaces are introduced to accommodate singularities and to measure the regularity of the eigenfunctions. Optimal error estimates for the mortar spectral element method and the lifting theorem for the eigenfunctions are established.

A [formula omitted] mortar spectral element method for the [formula omitted]-Laplacian equation

We present a constrained-vertex variant of the mortar spectral element method to solve the p-Laplacian equation. To show reliability of the method, first we investigate convergence rate of h-version and k-version for a sufficiently smooth solution. Then for solutions with limited regularity, we use a hk-strategy in which the size of elements is geometrically reduced towards the singularity while the polynomial degree is linearly reduced. In fact, we numerically study convergence rate of the hk mortar spectral element method in the L2-norm and H1-norm using geometrical meshes with elements of non-uniform polynomial degree. In this regard, we consider several benchmarks with various choices of the parameter p as well as geometric grading factors. To deal with possible degeneracy in the p-Laplacian operator, we add artificial diffusion to the original equation and then study the effects of this extra diffusion on convergence rate. To find the solution of highly nonlinear system arising from discretization we use an optimization technique based on the trust region method in which an analytical Jacobian matrix is utilized.

Solving the singular two-dimensional fourth order problem by the mortar spectral element method

In this work, we implement the mortar spectral element method for the biharmonic problem with a homogeneous boundary condition. We consider a polygonal domain with corners which relies on the mortar decomposition domain technique. We propose the Strang and Fix algorithm, which permits to enlarge the discrete space of the solution by the first singular function. The interest of this algorithm is the approximation of the solution and the leading singular coefficient which has a physical significance in the propagation of cracks. We give some numerical results which confirm the optimality of the order of the error.

3维Mortar谱元法模拟Kelvin-Helmhtz不稳定性混合层

3维Mortar谱元法模拟Kelvin-Helmhtz不稳定性混合层

A Numerical Study of Temporal Mixing Layer with Three-Dimensional Mortar Spectral Element Method

We investigate a set of well-posed interface conditions of viscous fluxes of Navier-Stokes equations, extend them to three-dimensional Navier-Stokes equations and successfully study a temporally turbulent mixing layer. The evolution of a large vortex structure and the non-dimensional statistical quantities in the mixing layer are numerically analyzed. The span-wise energy spectrum of the mixing layer exhibits wide band features, which means that the simulation generates the desired amount of small scales. These results show that too many small scales are present due to insufficient dissipation and the amount of large scales is low, indicating too much dissipation for large-scale structures.

The mortar spectral element method in domains of operators. Part II: the curl operator and the vector potential problem

The mortar spectral element method is a domain decomposition technique that allows for discretizing second- or fourth-order elliptic equations when set in standard Sobolev spaces. The aim of this paper is to extend this method to problems formulated in the space of square-integrable vector fields with square-integrable curl. We consider the problem of computing the vector potential associated with a divergence-free function in 3D and propose a discretization of it. The numerical analysis of the discrete problem is performed and numerical experiments are presented; they turn out to be in good agreement with the theoretical results.

Identification and Reconstruction of a Small Leak Zone in a Pipe by a Spectral Element Method

This paper deals with the identification of a zone permitting fluid to leak out of a drain. Using the analogy with crack identification by boundary measurements, we give uniqueness and stability results and we propose an algorithm for the numerical solution of the problem. Since the forward problem consists of a mixed boundary value problem with a strong anisotropy, we discretize it by a mortar spectral element method.

The mortar spectral element method in domains of operators. Part I: The divergence operator and Darcy's equations

The mortar spectral element method is a domain decomposition technique that allows for discretizing second- or fourth-order elliptic equations when set in standard Sobolev spaces. The aim of this paper is to extend this method to some problems formulated in spaces of square-integrable functions with square-integrable divergence. A discretization of the equations due to Darcy which model the flow in porous media is proposed. The numerical analysis of the discrete problem is performed and 2D numerical experiments are presented; they turn out to be in good agreement with the theoretical results. Résumé: La méthode d'éléments spectraux avec joints est une technique de décomposition de domaine permettant de discrétiser des équations elliptiques d'ordre 2 ou 4 posés dans des espaces de Sobolev usuels. Le but de cet article est d'étendre cette méthode à certains problèmes variationnels formulés dans des espaces de fonctions de carré intégrable à divergence de carré intégrable. On propose une discrétisation des équations de Darcy qui modélisent l'écoulement dans des milieux poreux, on en effectue l'analyse numérique et on présente des expériences numériques en dimension 2 qui s'avèrent cohérentes avec les résultats de l'analyse.

A mortar spectral element method for 3D Maxwell's equations

In this paper we propose a mortar spectral element method for solving Maxwell’s equations in 3D bounded cavities. The method is based on a non-conforming decomposition of the domain into the union of non-overlapping parallelepipeds. After proving an error estimate, we present some 3D computational results which confirm the performance of the method.

Handling Geometric Singularities by the Mortar Spectral Element Method I. Case of the Laplace Equation

The solution of an elliptic partial differential equation in a polygon is generally not regular. But it can be written as a sum of a regular part and a linear combination of singular functions. The purpose of our work is the numerical analysis and the implementation of the Strang and Fix algorithm, with the mortar spectral element method. We also compute with a high accuracy the coefficient of the leading singularity.

L'équation de Laplace avec conditions aux limites discontinues : convergence d'une discrétisation par éléments spectraux

We prove an existence result for the Laplace equation in a disk with discontinuous Dirichlet boundary conditions: 0 on part of the boundary and 1 on its complement. The problem is discretized by the mortar spectral element method and a convergence estimate is derived which is confirmed numerically.

Handling corner singularities by the mortar spectral element method

The solution of an elliptic partial differential equation in a polygon can be written as a linear combination of singular functions, which only depend on the geometry of the domain, and of a more regular part. The aim of this paper is double: presenting an efficient algorithm to approximate the solution despite the singularities, computing a high accuracy approximation of the coefficient of the leading singularity. Both results rely on the mortar spectral element method.