On the Effective Implementation and Capabilities of the Least-Squares Collocation Method for Solving Second-Order Elliptic Equations

Исследованы возможности численного метода коллокации и наименьших квадратов (КНК) на примерах кусочно-полиномиального решения задачи Дирихле для уравнений Пуассона и типа диффузии-конвекции с особенностями в виде больших градиентов и разрыва решения на границах раздела двух подобластей. Предложены и реализованы новые hp-варианты метода КНК, основанные на присоединении внутри области малых и/или вытянутых нерегулярных ячеек, отсекаемых криволинейной границей раздела от исходных прямоугольных ячеек сетки, к соседним самостоятельным ячейкам. Выписываются с учетом особенности условия согласования между собой кусков решения в ячейках, примыкающих с разных сторон к границе раздела. Проведено сравнение результатов, полученных методом КНК и другими высокоточными методами. Показаны преимущества и достоинства метода КНК. Для ускорения итерационного процесса применены современные алгоритмы и методы: предобуславливание; свойства локальной системы координат в методе КНК; ускорение, основанное на подпространствах Крылова; операция продолжения на многосеточном комплексе; распараллеливание. Исследовано влияние этих способов на количество итераций и время расчетов при аппроксимации полиномами различных степеней. The capabilities of the numerical least-squares collocation (LSC) method of the piecewise polynomial solution of the Dirichlet problem for the Poisson and diffusion-convection equations are investigated. Examples of problems with singularities such as large gradients and discontinuity of the solution at interfaces between two subdomains are considered. New hp-versions of the LSC method based on the merging of small and/or elongated irregular cells to neighboring independent cells inside the domain are proposed and implemented. They cut off by a curvilinear interface from the original rectangular grid cells. Taking into account the problem singularity the matching conditions between the pieces of the solution in cells adjacent from different sides to the interface are written out. The results obtained by the LSC method are compared with other high-accuracy methods. Advantages of the LSC method are shown. For acceleration of an iterative process modern algorithms and methods are applied: preconditioning, properties of the local coordinate system in the LSC method, Krylov subspaces; prolongation operation on a multigrid complex; parallelization. The influence of these methods on iteration numbers and computation time at approximation by polynomials of various degrees is investigated.

PREDICTION OF COMPRESSION AND EXTENSION ZONES IN GEOLOGICAL STRUCTURES BASED ONLY ON THE VELOCITIES OF LONGITUDINAL WAVES IN THE GEOLOGICAL MEDIUM

The article presents accurate solutions for the problem for two elastic half‐spaces with an arbitrary curvilinear interface. Our study shows that dilatation solutions (Poisson integrals) are dependent on neither an overall compression modulus nor the Poisson ratio, and depend only on the velocity of longitudinal waves. These specific solutions can be supplemented by general solutions for an incompressible elastic medium, and the boundary conditions of the rigid contact for the sum of the solutions can thus be satisfied. Relatively simple calculations make it possible to determine the divergence of the displacement field and reduce the entire problem solving process to a study of Poisson equations with a known divergence. Furthermore, predictions of volumetric compression or extension are important for geological investigations, since the zones characterized by reduced pressure rates may act as fluid attractors.