This paper analyzes a method for solving the thirdand fifth-order differential equations with constant coefficients using a Jacobi dual-Petrov–Galerkin method, which is more reasonable than the standard Galerkin one. The spatial approximation is based on Jacobi polynomials P (α,β) n with α, β ∈ (−1, ∞) and n is the polynomial degree. By choosing appropriate base functions, the resulting system ...

In the earlier paper [6] a Galerkin method was proposed and analyzed for the numerical solution of a Dirichlet problem for a semi-linear elliptic boundary value problem of the form U = F ( ; U). This was converted to a problem on a standard domain and then converted to an equivalent integral equation. Galerkin’s method was used to solve the integral equation, with the eigenfunctions of the Lapl...

In this work we design hybrid continuous-discontinuous finite element spaces that permit discontinuities on non-matching element interfaces of non-conforming meshes. Then, we develop an equal-order stabilized finite element formulation for incompressible flows over these hybrid spaces, which combines the element interior stabilization of SUPGtype continuous Galerkin formulations and the jump st...

Conventional high-order discontinuous Galerkin schemes suffer from interface errors caused by the misalignment between straight-sided elements and curved material interfaces. We develop a novel discontinuous Galerkin scheme to reduce the errors. Our new scheme use the correct normal vectors to the curved interfaces, while the conventional scheme uses the normal vectors to the element edge. We m...

In this article we study the properties of the hyperinterpolation operator on the unit disk D in R. We show how the hyperinterpolation can be used in connection with the Kumar-Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disk (discrete Galerkin method). A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results prove...

We propose and analyze a finite element method for a semi– stationary Stokes system modeling compressible fluid flow subject to a Navier– slip boundary condition. The velocity (momentum) equation is approximated by a mixed finite element method using the lowest order Nédélec spaces of the first kind. The continuity equation is approximated by a standard piecewise constant upwind discontinuous G...

Countable systems of ordinary differential equations appear frequently in chemistry, physics, biology and statistics. They can be considered as ordinary differential equations in sequence spaces. In this work, a fully adaptive algorithm for the computational treatment of such systems is developed. The method is based on a time discretization of an abstract Cauchy problem in Hubert space and a d...

We deal with the numerical solution of time-dependent convection-diffusion-reaction equations. We combine the local projection stabilization method for the space discretization with two different time discretization schemes: the continuous Galerkin-Petrov (cGP) method and the discontinuous Galerkin (dG) method of polynomial of degree k. We establish the optimal error estimates and present numer...

In uncertainty quantification, critical parameters of mathematical models are substituted by random variables. We consider dynamical systems composed of ordinary differential equations. The unknown solution is expanded into an orthogonal basis of the random space, e.g., the polynomial chaos expansions. A Galerkin method yields a numerical solution of the stochastic model. In the linear case, th...

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Urysohn integral equation. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, O(n−r) in L2-norm and O(n 1 2 −r) in infinity norm, and the iterated Legendre Galerkin ...