A fully discrete H 1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

Abstract

We propose and analyze a fully discrete H 1 -Galerkin method with quadrature for nonlinear parabolic advection-diffusion-reaction equations that requires only linear algebraic solvers. Our scheme applied to the special case heat equation is a fully discrete quadrature version of the least-squares method. We prove second order convergence in time and optimal H 1 convergence in space for the computer implementable method. The results of numerical computations demonstrate optimal order convergence of scheme in H k for k = 0, 1, 2.

A pseudospectral quadrature method for Navier-Stokes equations on rotating spheres

“…for x ∈ S and t ≥ 0, the standard fixed-time-step backward-Euler (or Crank-Nicolson) Galerkin approach could be used in (4.1), leading to a first-order (or a second-order, respectively) in time non-adaptive scheme [13]. However, due to the complicated unknown flow behavior of the NSE solutions, when the initial states are random, it is more efficient instead to integrate (4.1) using a combination of multi-order integration formulas that allow adaptive choice of time step, leading to computation of solutions with a specified accuracy in time.…”

Schrödinger eigenbasis on a class of superconducting surfaces: Ansatz, analysis, FEM approximations and computations

“…The main aim of this section is to prove that: (i) for each winding number L ∈ Z, there is a sequence of non-negative eigenvalues λ j,L , j ∈ N, of the weak-form Schrödinger operator S on the C 2 manifold S; (ii) the ansatz of the associated functions can be written in the separated coordinate form χ j,L (s)e iLθ , j ∈ N, L ∈ Z; (10) (iii) the eigenfunctions are in the null-space of B λ j,L (with j ∈ N, L ∈ Z); and (iv) the eigenfunctions constitute a complete orthogonal system for L 2 (S; C). In this process, we identify an arbitrarily near approximate spectral problem for the unknown factors in (10).…”

“…(This FEM dimension may be reduced by requiring higher continuity of the spline basis with associated restriction on r and the Galerkin procedure below may be replaced with appropriate Petrov-Galerkin scheme [10,12].) Our discrete formulation of (26) is then to find,…”

Continuous and discrete time Zhang dynamics for time-varying 4th root finding

“…Online solution of 4th root in the form of x 4 − a = 0, where a is a positive scalar, is considered to be an important special case of nonlinear-equation problem arising in science and engineering fields [1][2][3][4][5][6][7][8]; e.g., in [1], one image could be described in Torelli group by solving its fourth root of unity; in [2], Harris corner strength image may be refined from the fourth root of original image. Thus, many related numerical algorithms are presented for such a problem solving [3,[5][6][7][8].…”