On the mesh nonsingularity of the moving mesh PDE method

The moving mesh PDE (MMPDE) method for variational mesh generation and adaptation is studied theoretically at the discrete level, in particular the nonsingularity of the obtained meshes. Meshing functionals are discretized geometrically and the MMPDE is formulated as a modified gradient system of the corresponding discrete functionals for the location of mesh vertices. It is shown that if the meshing functional satisfies a coercivity condition, then the mesh of the semi-discrete MMPDE is nonsingular for all time if it is nonsingular initially. Moreover, the altitudes and volumes of its elements are bounded below by positive numbers depending only on the number of elements, the metric tensor, and the initial mesh. Furthermore, the value of the discrete meshing functional is convergent as time increases, which can be used as a stopping criterion in computation. Finally, the mesh trajectory has limiting meshes which are critical points of the discrete functional. The convergence of the mesh trajectory can be guaranteed when a stronger condition is placed on the meshing functional. Two meshing functionals based on alignment and equidistribution are known to satisfy the coercivity condition. The results also hold for fully discrete systems of the MMPDE provided that the time step is sufficiently small and a numerical scheme preserving the property of monotonically decreasing energy is used for the temporal discretization of the semi-discrete MMPDE. Numerical examples are presented.Comment: Revised and improved version of the WIAS preprin

Numerical investigations of traveling singular sources problems via moving mesh method

This paper studies the numerical solution of traveling singular sources problems. In such problems, a big challenge is the sources move with different speeds, which are described by some ordinary differential equations. A predictor-corrector algorithm is presented to simulate the position of singular sources. Then a moving mesh method in conjunction with domain decomposition is derived for the underlying PDE. According to the positions of the sources, the whole domain is splitted into several subdomains, where moving mesh equations are solved respectively. On the resulting mesh, the computation of jump $[\dot{u}]$ is avoided and the discretization of the underlying PDE is reduced into only two cases. In addition, the new method has a desired second-order of the spatial convergence. Numerical examples are presented to illustrate the convergence rates and the efficiency of the method. Blow-up phenomenon is also investigated for various motions of the sources

On the mesh nonsingularity of the moving mesh PDE method

The moving mesh PDE (MMPDE) method for variational mesh generation and adaptation is studied theoretically at the discrete level, in particular the nonsingularity of the obtained meshes. Meshing functionals are discretized geometrically and the MMPDE is formulated as a modified gradient system of the corresponding discrete functionals for the location of mesh vertices. It is shown that if the meshing functional satisfies a coercivity condition, then the mesh of the semi-discrete MMPDE is nonsingular for all time if it is nonsingular initially. Moreover, the altitudes and volumes of its elements are bounded below by positive numbers depending only on the number of elements, the metric tensor, and the initial mesh. Furthermore, the value of the discrete meshing functional is convergent as time increases, which can be used as a stopping criterion in computation. Finally, the mesh trajectory has limiting meshes which are critical points of the discrete functional. The convergence of the mesh trajectory can be guaranteed when a stronger condition is placed on the meshing functional. Two meshing functionals based on alignment and equidistribution are known to satisfy the coercivity condition. The results also hold for fully discrete systems of the MMPDE provided that the time step is sufficiently small and a numerical scheme preserving the property of monotonically decreasing energy is used for the temporal discretization of the semi-discrete MMPDE. Numerical examples are presente

Surface and bulk moving mesh methods based on equidistribution and alignment

In this dissertation, we first present a new functional for variational mesh generation and adaptation that is formulated by combining the equidistribution and alignment conditions into a single condition with only one dimensionless parameter. The functional is shown to be coercive which, when employed with the moving mesh partial differential equation method, allows various theoretical properties to be proved. Numerical examples for bulk meshes demonstrate that the new functional performs comparably to a similar existing functional that is known to work well but contains an additional parameter. Variational mesh adaptation for bulk meshes has been well developed however, surface moving mesh methods are limited. Here, we present a surface moving mesh method for general surfaces with or without explicit parameterization. The development starts with formulating the equidistribution and alignment conditions for surface meshes from which, we establish a meshing energy functional. The moving mesh equation is then defined as the gradient system of the energy functional, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. A selection of two-dimensional and three-dimensional examples are presented

Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters

Preprint version, the final publication is available at Springer via http://dx.doi.org/10.1007/s10543-015-0559-8This paper discusses the numerical solution of 1-D convection-diffusion-reaction problems that are singularly perturbed with two small parameters using a new mesh-adaptive upwind scheme that adapts to the boundary layers. The meshes are generated by the equidistribution of a special positive monitor function. Uniform, parameter independent convergence is shown and holds even in the limit that the small parameters are zero. Numerical experiments are presented that illustrate the theoretical findings, and show that the new approach has better accuracy compared with current methods.DFG, SFB 1029, Substantial efficiency increase in gas turbines through direct use of coupled unsteady combustion and flow dynamic

Stochastic domain decomposition for time dependent adaptive mesh generation

The efficient generation of meshes is an important component in the numerical solution of problems in physics and engineering. Of interest are situations where global mesh quality and a tight coupling to the solution of the physical partial differential equation (PDE) is important. We consider parabolic PDE mesh generation and present a method for the construction of adaptive meshes in two spatial dimensions using stochastic domain decomposition that is suitable for an implementation in a multi– or many–core environment. Methods for mesh generation on periodic domains are also provided. The mesh generator is coupled to a time dependent physical PDE and the system is evolved using an alternating solution procedure. The method uses the stochastic representation of the exact solution of a parabolic linear mesh generator to find the location of an adaptive mesh along the (artificial) subdomain interfaces. The deterministic evaluation of the mesh over each subdomain can then be obtained completely independently using the probabilistically computed solutions as boundary conditions. The parallel performance of this general stochastic domain decomposition approach has previously been shown. We demonstrate the approach numerically for the mesh generation context and compare the mesh obtained with the corresponding single domain mesh using a representative mesh quality measure

Domain decomposition approaches for the generation of equidistributing parametric curves

Moving mesh methods are often used to solve boundary value problems whose solutions contain regions of rapid change. In this case, these moving mesh methods allow us to concentrate a fixed number of nodes in these regions of high variance. These meshes are obtained by solving a second order boundary value problem (BVP), which arises from the equidistribution principle. There are many real-world examples where boundary value problems are posed on curves and surfaces. Here, we focus on the case where the problem is posed on a curve that is able to be explicitly represented parametrically as x = (x₁(r), x₂(r), xₙ(r) ℝⁿ. When the solution has regions of rapid change or the curve on which the problem is posed has regions of high variance or curvature, moving mesh methods allow us to find a mesh that better resolves the function on the curve without adding additional nodes. We consider combining the solution of mesh equations with the solution of differential equations posed on parametric curves. These differential equations include both time-dependent partial differential equations (PDEs) and time-independent boundary layer problems. In addition to considering the above on a single domain, we extend these methods to form multi-domain iterations to solve these boundary value problems. Domain decomposition allows us to harness the power of parallel computing, a topic that has become popular in recent years with the increase of computing power. We provide multi-domain iterations for both time-dependent and time-independent differential equations posed on parametric curves; these include classical Schwarz and optimized Schwarz methods. These iterations are formed such that they are able to be performed in parallel. Numerical results are provided throughout to illustrate the results of the iterations. This thesis also includes theoretical results that generalize known results for classical Schwarz and optimized Schwarz methods to the case where the problem is defined on a curve