We present some recent advances in the productive and symbiotic interplay between general potential theories (subharmonic functions associated to closed subsets F⊂J2(X) of the 2-jets on X⊂Rn open) ...and subsolutions of degenerate elliptic and parabolic PDEs of the form F(x,u,Du,D2u)=0. We will implement the monotonicity-duality method begun by Harvey and Lawson (2009) (in the pure second order constant coefficient case) for proving comparison principles for potential theories where F has sufficient monotonicity and fiberegularity (in variable coefficient settings) and which carry over to all differential operators F which are compatible with F in a precise sense for which the correspondence principle holds.
We will consider both elliptic and parabolic versions of the comparison principle in which the effect of boundary data is seen on the entire boundary or merely on a proper subset of the boundary.
Particular attention will be given to gradient dependent examples with the requisite sufficient monotonicity of proper ellipticity and directionality in the gradient. Examples operators we will discuss include the degenerate elliptic operators of optimal transport in which the target density is strictly increasing in some directions as well as operators which are weakly parabolic in the sense of Krylov. Further examples, modeled on hyperbolic polynomials in the sense of Gårding give a rich class of examples with directionality in the gradient. Moreover we present a model example in which the comparison principle holds, but standard viscosity structural conditions fail to hold.
In this work, a new cubic B-spline-based semi-analytical algorithm is presented for solving 3D anisotropic convection-diffusion-reaction (CDR) problems in the inhomogeneous medium. The mathematical ...model is expressed by the quasi-linear second-order elliptic partial differential equations (EPDE) with mixed derivatives and variable coefficients. The final approximation is obtained as a sum of the rough primary solution and the modified spline interpolants with free parameters. The primary solution mathematically satisfies boundary conditions. Thus, the free parameters of interpolants are chosen to satisfy the governing equation in the solution domain. The numerical examples demonstrate the high accuracy of the proposed method in solving 3D CDR problems in single- and multi-connected domains.
We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear isotropic elliptic equations on compact Riemannian manifolds, depending ...only on dimension and a lower bound for the Ricci curvature. These estimates imply sharp gradient bounds relating the gradient of an arbitrary solution at given height to that of a symmetric solution on a warped product model space. We also discuss the problem on Finsler manifolds with nonnegative weighted Ricci curvature, and on complete manifolds with bounded geometry, including solutions on manifolds with boundary with Dirichlet boundary condition. Particular cases of our results include gradient estimates of Modica type.
This paper presents a new technique based on a collocation method using cubic splines for second order elliptic equation with irregularities in one dimension and two dimensions. The differential ...equation is first collocated at the two smooth sub domains divided by the interface. We extend the sub domains from the interior of the domain and then the scheme at the interface is developed by patching them up. The scheme obtained gives the second order accurate solution at the interface as well as at the regular points. Second order accuracy for the approximations of the first order and the second order derivative of the solution can also be seen from the experiments performed. Numerical experiments for 2D problems also demonstrate the second order accuracy of the present scheme for the solution u and the derivatives ux,uxx and the mixed derivative uxy. The approach to derive the interface relations, established in this paper for elliptic interface problems, can be helpful to derive high order accurate numerical methods. Numerical tests exhibit the super convergent properties of the scheme.
•A fourth-order kernel-free boundary integral method in three space dimensions.•Representation of irregular surfaces by intersection points with a Cartesian grid.•A 27-point compact finite difference ...scheme on irregular domains.
The kernel-free boundary integral (KFBI) method is a finite difference version of the traditional boundary integral method for elliptic and parabolic partial differential equations on complex domains. It evaluates boundary or volume integrals involved in the solution of boundary integral equations (BIEs) by solving equivalent but simple interface problems on regular grids, so that the integral kernel or Green function is never needed or computed. This is the essential difference of the KFBI method from the traditional ones. It takes advantage of the well-conditioning property of discrete BIEs so that the number of Krylov subspace iterations is essentially independent of discretization parameter or system dimension. This paper presents a fourth-order kernel-free boundary integral method for second-order elliptic partial differential equations on complex domains in three space dimensions, whose boundaries are given by implicitly defined surfaces. It represents the domain boundary and discretizes data on it by intersection points of the surface with Cartesian grid lines. The approach has a variety of advantages. The current work solves simple interface problems with corrected 27-point compact finite difference schemes and calculates the discrete equations with a fast Fourier transform based elliptic solver. Numerical examples show that the proposed method is efficient as well as accurate.
At present, the quality of online video courses in China is mixed. There are several reasons for the quality of online video courses. 1. The advantages and disadvantages of the front-end video ...capture equipment itself; 2. The distance of online video transmission; 3. The medium through which the video is transmitted; 4. Watch whether there is relevant interference information in the signal where the video is located and whether the video is compressed during transmission. These reasons lead to that although there is much to learn in the video, the resolution is too low to see from the video. With the development of the current social environment, most of the courses need online teaching. Therefore, in order to improve some problems in video playing caused by the increase of online teaching amount caused by the current environment, this paper provides higher resolution video for online courses by using high-resolution image processing technology based on the elliptic partial differential equation online video course. The high resolution processing technology used in this paper is centered on filtering algorithm. On the basis of the existing online video course of elliptic partial differential equations, the use of high-resolution technology can overcome the resolution limit of the hardware itself and further improve the video quality of online video teaching.
By a probabilistic method, we prove the existence and uniqueness of weak solutions to Neumann problems for a class of semi-linear elliptic partial differential equations with nonlinear singular ...divergence terms, which can only be understood in distributional sense. This leads to the further study on a new class of infinite horizon backward stochastic differential equations, which involves integrals with respect to a forward–backward martingale and a singular continuous increasing process.
•Extends the applicability of boundary integral methods to inhomogeneous PDEs.•A simple to implement and efficient algorithm for numerical function extension.•The most computationally costly part can ...be precomputed and reused.•The extended function has compact support and user specified regularity.•Solution to the Poisson equation converges as a tenth order method to round off.
We introduce an efficient algorithm, called partition of unity extension or PUX, to construct an extension of desired regularity of a function given on a complex multiply connected domain in 2D. Function extension plays a fundamental role in extending the applicability of boundary integral methods to inhomogeneous partial differential equations with embedded domain techniques. Overlapping partitions are placed along the boundaries, and a local extension of the function is computed on each patch using smooth radial basis functions; a trivially parallel process. A partition of unity method blends the local extrapolations into a global one, where weight functions impose compact support. The regularity of the extended function can be controlled by the construction of the partition of unity function. We evaluate the performance of the PUX method in the context of solving the Poisson equation on multiply connected domains using a boundary integral method and a spectral solver. With a suitable choice of parameters the error converges as a tenth order method down to 10−14.
This study compares various multigrid strategies for the fast solution of elliptic equations discretized by the hybrid high‐order method. Combinations of h$$ h $$‐, p$$ p $$‐, and hp$$ hp ...$$‐coarsening strategies are considered, combined with diverse intergrid transfer operators. Comparisons are made experimentally on 2D and 3D test cases, with structured and unstructured meshes, and with nested and non‐nested hierarchies. Advantages and drawbacks of each strategy are discussed for each case to establish simplified guidelines for the optimization of the time to solution.
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