An hp-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems

In this paper we develop an hp-adaptive procedure for the numerical solution of general second-order semilinear elliptic boundary value problems, with possible singular perturbation. Our approach combines both adaptive Newton schemes and an hp-version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust hp-version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples

Goal-oriented error analysis of iterative Galerkin discretizations for nonlinear problems including linearization and algebraic errors

We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a suitable linearization which guarantees the adjoint consistency of the numerical scheme. We derive error estimates and develop an efficient adaptive algorithm which balances the errors arising from the discretization and use of iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.Comment: submitte

Optimal Control of Convective FitzHugh-Nagumo Equation

We investigate smooth and sparse optimal control problems for convective FitzHugh-Nagumo equation with travelling wave solutions in moving excitable media. The cost function includes distributed space-time and terminal observations or targets. The state and adjoint equations are discretized in space by symmetric interior point Galerkin (SIPG) method and by backward Euler method in time. Several numerical results are presented for the control of the travelling waves. We also show numerically the validity of the second order optimality conditions for the local solutions of the sparse optimal control problem for vanishing Tikhonov regularization parameter. Further, we estimate the distance between the discrete control and associated local optima numerically by the help of the perturbation method and the smallest eigenvalue of the reduced Hessian

An adaptive space-time Newton–Galerkin approach for semilinear singularly perturbed parabolic evolution equations

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)In this article, we develop an adaptive procedure for the numerical solution of semilinear parabolic problems with possible singular perturbations. Our approach combines a linearization technique using Newton’s method with an adaptive discretization – which is based on a spatial finite element method and the backward Euler time-stepping scheme – of the resulting sequence of linear problems. Upon deriving a robust a posteriori error analysis, we design a fully adaptive Newton-Galerkin time-stepping algorithm. Numerical experiments underline the robustness and reliability of the proposed approach for various examples

Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs

In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space V_{H,P}. The resulting `coarse' numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space V_{h,p}; thereby, only a linear system of equations is solved on the richer space V_{h,p}. In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces V_{H,P} and V_{h,p}, respectively, in an automatic fashion. Numerical experiments are presented for both two- and three-dimensional problems; in each case, we demonstrate that the cpu time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method

An adaptive space-time Newton–Galerkin approach for semilinear singularly perturbed parabolic evolution equations

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)In this article, we develop an adaptive procedure for the numerical solution of semilinear parabolic problems with possible singular perturbations. Our approach combines a linearization technique using Newton’s method with an adaptive discretization – which is based on a spatial finite element method and the backward Euler time-stepping scheme – of the resulting sequence of linear problems. Upon deriving a robust a posteriori error analysis, we design a fully adaptive Newton-Galerkin time-stepping algorithm. Numerical experiments underline the robustness and reliability of the proposed approach for various examples

Two-grid hp-version discontinuous Galerkin finite element methods for quasi-Newtonian fluid flows

In this article we consider the a priori and a posteriori error analysis of two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a strongly monotone quasi-Newtonian fluid flow problem. The basis of the two-grid method is to first solve the underlying nonlinear problem on a coarse finite element space; a fine grid solution is then computed based on undertaking a suitable linearization of the discrete problem. Here, we study two alternative linearization techniques: the first approach involves evaluating the nonlinear viscosity coefficient using the coarse grid solution, while the second method utilizes an incomplete Newton iteration technique. Energy norm error bounds are deduced for both approaches. Moreover, we design an hp-adaptive refinement strategy in order to automatically design the underlying coarse and fine finite element spaces. Numerical experiments are presented which demonstrate the practical performance of both two-grid discontinuous Galerkin methods

Gradient Flow Finite Element Discretizations with Energy-Based Adaptivity for the Gross-Pitaevskii Equation

We present an effective adaptive procedure for the numerical approximation of the steady-state Gross-Pitaevskii equation. Our approach is solely based on energy minimization, and consists of a combination of gradient flow iterations and adaptive finite element mesh refinements. Numerical tests show that this strategy is able to provide highly accurate results, with optimal convergence rates with respect to the number of freedom