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Partial differential equations (PDEs) with spatial derivatives of order higher than two are receiving increasing attention, partially due to the current popularity of the phase-field method. In the finite element community, the growth of Isogeometric Analysis and other discretization schemes that employ approximation spaces with high-order continuity has fostered the use of variational formulations that avoid the use of auxiliary unknowns representing derivatives of the solution. However, one of the caveats of this approach is the accurate and efficient imposition of boundary conditions, especially on complex geometries. This paper proposes a new method to impose boundary conditions naturally in the weak form of higher-order PDEs. Our method is based on a specially-designed weak form in which the boundary conditions to be imposed are weighted by derivatives of the weight functions. This requires multiple integrations by parts which are allowed due to the smoothness of the basis functions. The Cahn–Hilliard equation and the isothermal Navier–Stokes–Norteweg equations are used as examples to demonstrate the proposed method. We show that if the solutions of the PDEs are smooth enough, the proposed variational equations and the original PDEs are equivalent. We discretize the variational equations using Isogeometric Analysis. Convergence results of the proposed method agree with the best approximation errors of the basis functions. Numerical examples illustrate the applicability of the approach to mapped geometries with non-conformal grids. [ABSTRACT FROM AUTHOR]

This work focuses on the development of a posteriori error estimates for fourth-order, elliptic, partial differential equations. In particular, we propose a novel algorithm to steer an adaptive simulation in the context of Kirchhoff plates and Kirchhoff–Love shells by exploiting the local refinement capabilities of hierarchical B-splines. The method is based on the solution of an auxiliary residual-like variational problem, formulated by means of a space of localized spline functions. This space is characterized by C 1 continuous B-splines with compact support on each active element of the hierarchical mesh. We demonstrate the applicability of the proposed estimator to Kirchhoff plates and Kirchhoff–Love shells by studying several benchmark problems which exhibit both smooth and singular solutions. In all cases, we obtain optimal asymptotic rates of convergence for the error measured in the energy norm and an excellent approximation of the true error. • Development of a novel estimator for isogeometric Kirchhoff plates and shells. • Based on a small, easily-invertible auxiliary problem. • Avoid the computation of high-order derivatives. • Adaptive strategy recovers optimal rates of convergence. • Excellent estimation of the true error. [ABSTRACT FROM AUTHOR]

In this work we extend our recently proposed adaptive refinement strategy for h p -finite element approximations of elliptic problems by taking into account an inexact algebraic solver. Namely, on each level of refinement and on each iteration of an (arbitrary) iterative algebraic solver, we compute guaranteed a posteriori error bounds on the algebraic and the total errors in energy norm. The algebraic error is the difference between the inexact discrete solution obtained by an iterative algebraic solver and the (unavailable) exact discrete solution. On the other hand, the total error stands for the difference between the inexact discrete solution and the (unavailable) exact solution of the partial differential equation. For the algebraic error upper bound, we crucially exploit the whole nested hierarchy of h p -finite element spaces created by the adaptive algorithm, whereas the remaining parts of the total error upper and lower bounds are computed using the finest space only. These error bounds allow us to formulate adaptive stopping criteria for the algebraic solver ensuring that the algebraic error does not significantly contribute to the total error. Next, we use the total error bound to mark mesh vertices for refinement via Dörfler's bulk-chasing criterion. On patches associated with marked vertices only, we solve two separate primal finite element problems with homogeneous Dirichlet (Neumann) boundary conditions, which serve to decide between h -, p -, or h p -refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the ratio of the total errors of the inexact approximations between successive refinements (the error reduction factor), when the stopping criteria are satisfied. Finally, in a series of numerical experiments, we investigate the practicality of the proposed adaptive solver, the accuracy of our bound on the reduction factor, and show that exponential convergence rates are achieved even in the presence of an inexact algebraic solver. • Adaptive h p -refinement strategy taking into account an inexact algebraic solver. • Computable guaranteed bound on the ratio of errors between successive refinements. • Adaptive stopping criteria for the algebraic solver on each level of refinement. • Numerics: exponential convergence rates in presence of an inexact algebraic solver. [ABSTRACT FROM AUTHOR]