Wavelet Numerical Solutions for a Class of Elliptic Equations with Homogeneous Boundary Conditions

This paper focuses on a method to construct wavelet Riesz bases with homogeneous boundary condition and use them to a kind of second-order elliptic equation. First, we construct the splines on the interval [0,1] and consider their approximation properties. Then we define the wavelet bases and illustrate the condition numbers of stiffness matrices are small and bounded. Finally, several numerical examples show that our approach performs efficiently.

Boundary problem with integral displacement for the model equation of elliptic type

Для модельного эллиптического уравнения второго порядка рассматривается метод редукции нелокальных краевых задач с интегральным смещением к локальным краевым задачам для уравнения более высокого порядка составного типа. Исследуется разрешимость поставленных задач. For a model second order elliptic equation is considered the method of reduction of nonlocal boundary value problems with integral offset to the local boundary value problems for equations of higher order composite type. The solvability of tasks is investigated.

A High-Order HDG Method with Dubiner Basis for Elliptic Flow Problems

We propose a standard hybridizable discontinuous Galerkin (HDG) method to solve a classic problem in fluids mechanics: Darcy’s law. This model describes the behavior of a fluid trough a porous medium and it is relevant to the flow patterns on the macro scale. Here we present the theoretical results of existence and uniqueness of the weak and discontinuous solution of the second order elliptic equation, as well as the predicted convergence order of the HDG method. We highlight the use and implementation of Dubiner polynomial basis functions that allow us to develop a general and efficient high order numerical approximation. We also show some numerical examples that validate the theoretical results.