Adaptive pseudo‐transient‐continuation‐Galerkin methods for semilinear elliptic partial differential equations

Abstract

Abstract. In this paper we investigate the application of pseudo-transient-continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, employ the PTC-methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction-type PTC-method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.

A numerical energy minimisation approach for semilinear diffusion-reaction boundary value problems based on steady state iterations

“…for a suitable initial guess v 0 : Ω → R (with zero boundary values). An unpretentious way to discretise (2) with respect to time is the forward Euler scheme (with a time step ∆t > 0). It yields an iteratively generated sequence {u n } n that is obtained by solving the linear elliptic problem…”

“…In the present paper, we circumvent this issue by making use of the underlying variational framework associated to (1), and prove that the sequence {u n } n resulting from (3) features some favourable energy properties; these, in turn, allow to establish an alternative convergence analysis. We remark in passing that, instead of using the forward Euler method (or another explicit time marching scheme) for the approximation of steady-state solutions to (2), the backward Euler method could also be of interest in light of its unconditional stability. Evidently, this approach requires the application of a suitable nonlinear solver in each discrete temporal step.…”

“…Evidently, this approach requires the application of a suitable nonlinear solver in each discrete temporal step. For instance, combining the backward Euler discretisation with the Newton iteration scheme, the so-called pseudo-transient-continuation (PTC) method presented in [7] emerges; we also refer to [2], where the PTC approach was investigated in the specific context of semilinear singularly perturbed problems.…”

Adaptive fixed point iterations for semilinear elliptic partial differential equations

“…Similarly, in [1,2,4,12,13,17], the nonlinear PDE problems at hand are linearized by an (adaptive) Newton technique, and subsequently discretized by a linear finite element method. On a related note, the discretization of a sequence of linearized problems resulting from the local approximation of semilinear evolutionary problems has been investigated in [3]. In all of the works [1][2][3][4]9], the key idea in obtaining fully adaptive discretization schemes is to provide a suitable interplay between the underlying linearization procedure and (adaptive) Galerkin methods; this is based on investing computational time into whichever of these two aspects is currently dominant.…”

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

“…Nonlinear evolution equations are often used to describe some physical aspects that arise in the various fields of nonlinear sciences, such as plasma physics, quantum mechanics, biological sciences, chemistry, chemical physics, and so forth. Various powerful techniques have been formulated and used by different scholars to find the solutions of some NLEEs, such as the sine-Gordon expansion method [1][2][3], the generalized Kudryashov method [4,5], the extended tanh method [6,7], the new generalized and improved (G /G)-expansion method [8], the Jacobi elliptic function method [9,10], the improved Bernoulli subequation function method [11], the tanh method [12,13], the sine-cosine method [14], the Lie group analysis method [15][16][17], the homogeneous balance method [18], the modified simple equation method [19,20], the meshless method of radial basis functions [21], He's variational iteration method [22], the explicit multistep Galerkin finite element method [23], the differential quadrature based numerical method [24], the partitioned second-order method [25], the adaptive pseudo-transient-continuation-Galerkin methods [26]. In general, various efficient techniques have been implemented to explore the search for the solutions of the different kind of NLEEs [27][28][29][30][31][32][33][34][35][36][37][38].…”