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To address the issue of low accuracy resulting from approximation errors in conventional interval finite element methods, this study presents an ɛ -accelerated bivariate dimension-reduction interval finite element method. The proposed method aims to accurately and efficiently predict the static response of structures with high dimensionality and large uncertainty parameters. We first approximate the structural interval equilibrium equation using a bivariate dimension-reduction method. An explicit expression for the sequence of displacements is derived by approximating the inverse of the structural interval stiffness matrix using the Neumann series. Adjoint-based sensitivity analysis can effectively avoid the response interval expansion caused by parameter coupling. For improving the convergence speed and computational accuracy, the displacement response sequence is accelerated by a vector ɛ acceleration algorithm and an adaptive strategy. Finally, three engineering structures with uncertain-but-bounded parameters are analyzed with the proposed method. Compared with conventional interval analysis methods, the proposed method effectively balances computational accuracy and efficiency, making it more suitable for non-linear and large uncertainty problems with multiple interval parameters. The study shows that ɛ -accelerated bivariate dimension-reduction interval finite element method has a promising application. • An interval finite element method based on bivariate dimension-reduction is proposed. • Explicit sequences and errors for the bounds of displacements were derived. • Dependencies between parameters are addressed by adjoint-based sensitivity analysis. • Convergence speed and accuracy are improved by the adaptive ɛ algorithm. • Structural response intervals are efficiently and accurately predicted. [ABSTRACT FROM AUTHOR]

A simple flux reconstruction for finite element solutions of reaction–diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approximation of singularly perturbed reaction–diffusion problem. The flux reconstruction is based on simple, independent post-processing operations over patches of elements in conjunction with standard Raviart–Thomas vector fields. We prove that the corresponding local error indicators are locally efficient and robust (up to a nonstandard oscillation term) with respect to any mesh size and any size of the reaction coefficient, including the singularly perturbed limit. • A posteriori error bounds for singularly perturbed reaction–diffusion problems. • Proof of guaranteed upper bound on the error and robust local efficiency. • Fully computable estimator, fast to evaluate, and easily parallelizable. • Non-constant reaction coefficient, mixed boundary conditions, arbitrary dimension. [ABSTRACT FROM AUTHOR]

We propose a multi-component approach for improving the training of the physics-informed neural network (PINN) model for parabolic problems with a sharply perturbed initial condition. As an example of a parabolic problem, we consider the advection–dispersion equation (ADE) with a point (Gaussian) source initial condition. In the d -dimensional ADE, perturbations in the initial condition decay with time t as t − d / 2. We demonstrate that for d ≥ 2 , this decay rate can cause a large approximation error in the PINN solution. Furthermore, localized large gradients in the ADE solution make the (common in PINN) Latin hypercube sampling of the equation's residual highly inefficient. Finally, the PINN solution of parabolic equations is sensitive to the choice of weights in the loss function. We propose a normalized form of ADE where the initial perturbation of the solution does not decrease in amplitude and demonstrate that this normalization significantly reduces the PINN approximation error. Next, we present an adaptive sampling scheme based on the analytical estimate of the solution decay rate that significantly reduces the PINN estimation error for the same number of sampling (residual) points. Finally, we develop criteria for selecting weights based on the order of magnitude of different terms in the loss function. We demonstrate the accuracy of the proposed PINN model for forward, inverse, and backward ADEs. [ABSTRACT FROM AUTHOR]

Abstract Gaussian random fields are popular models for spatially varying uncertainties, arising for instance in geotechnical engineering, hydrology or image processing. A Gaussian random field is fully characterised by its mean function and covariance operator. In more complex models these can also be partially unknown. In this case we need to handle a family of Gaussian random fields indexed with hyperparameters. Sampling for a fixed configuration of hyperparameters is already very expensive due to the nonlocal nature of many classical covariance operators. Sampling from multiple configurations increases the total computational cost severely. In this report we employ parameterised Karhunen–Loève expansions for sampling. To reduce the cost we construct a reduced basis surrogate built from snapshots of Karhunen–Loève eigenvectors. In particular, we consider Matérn-type covariance operators with unknown correlation length and standard deviation. We suggest a linearisation of the covariance function and describe the associated online–offline decomposition. In numerical experiments we investigate the approximation error of the reduced eigenpairs. As an application we consider forward uncertainty propagation and Bayesian inversion with an elliptic partial differential equation where the logarithm of the diffusion coefficient is a parameterised Gaussian random field. In the Bayesian inverse problem we employ Markov chain Monte Carlo on the reduced space to generate samples from the posterior measure. All numerical experiments are conducted in 2D physical space, with non-separable covariance operators, and finite element grids with ∼ 1 0 4 degrees of freedom. [ABSTRACT FROM AUTHOR]

We derive a posteriori error estimators for an optimal control problem governed by a convection–reaction–diffusion equation; control constraints are also considered. We consider a family of low-order stabilized finite element methods to approximate the solutions to the state and adjoint equations. We obtain a fully computable a posteriori error estimator for the optimal control problem. All the constants that appear in the upper bound for the error are fully specified. Therefore, the proposed estimator can be used as a stopping criterion in adaptive algorithms. We also obtain a robust a posteriori error estimator for when the error is measured in a norm that involves the dual norm of the convective derivative. Numerical examples, in two and three dimensions, are presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

This paper introduces optimally-blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations. To quantify the approximation errors when we modify the inner products, we generalize the Pythagorean eigenvalue theorem of Strang and Fix. The proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non-uniform meshes as well as for different polynomial orders and continuity of the basis. The optimally-blended schemes improve the convergence rate of the method by two orders with respect to the fully-integrated Galerkin method. The proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel a posteriori error estimator for the conforming finite element approximation to the H ( curl ) problem with inhomogeneous media and with the right-hand side only in L 2 . The estimator is of the recovery type. Independent with the current approximation to the primary variable (the electric field), an auxiliary variable (the magnetizing field) is recovered in parallel by solving a similar H ( curl ) problem. An alternate way of recovery is presented as well by localizing of the error flux. The estimator is then defined as the sum of the modified element residual and the residual of the constitutive equation defining the auxiliary variable. It is proved that the estimator is approximately equal to the true error in the energy norm without the quasi-monotonicity assumption. Finally, we present numerical results for several H ( curl ) interface problems. [ABSTRACT FROM AUTHOR]

This article presents error bounds for a velocity–pressure segregated POD reduced order model discretization of the Navier–Stokes equations. The stability is proven in L ∞ (L 2) and energy norms for velocity, with bounds that do not depend on the viscosity, while for pressure it is proven in a semi-norm of the same asymptotic order as the L 2 norm with respect to the mesh size. The proposed estimates are calculated for the two flow problems, the flow past a cylinder and the lid-driven cavity flow. Their quality is then assessed in terms of the predicted logarithmic slope with respect to the velocity POD contribution ratio. We show that the proposed error estimates allow a good approximation of the real errors slopes and thus a good prediction of their rate of convergence. [ABSTRACT FROM AUTHOR]

In practical engineering problems, it is frequently challenging to collect sufficient data to construct high-precision probabilistic models. In this case, probabilistic models typically contain cognitive uncertainty and may be biased. Confidence-based design optimization (CBDO) is an effective tool for solving optimization problems with insufficient input data. Therein, epistemic uncertainty is taken into account, and its propagation is quantified by the confidence level of reliability. However, the coupling of reliability analysis and confidence analysis (CA) results in low calculation accuracy and an unacceptable rise in computational cost. To improve the performance of CBDO, a sequential single-loop reliability optimization and confidence analysis method (SROCA) is proposed in this paper. Specifically, a decoupling strategy with a serial of cycles of complete single-loop method (CSLM) and CA is established. In each cycle, CSLM and CA are decoupled from each other. Therein, the CSLM integrates both the reliability and confidence constraints into the deterministic optimization loop, and the performance functions are not required to be evaluated in CA due to the application of a semi-analytic sensitivity method. Additionally, update strategies for both the reliability and confidence indexes are employed to neutralize the linear approximation errors based on the dimension reduction method. Finally, four mathematical and one engineering example are adopted to demonstrate the performance of the proposed SROCA. The results show that the efficiency and accuracy of implementing CBDO achieve significant improvement by the proposed method. [ABSTRACT FROM AUTHOR]

In this work we propose and analyze an HDG method for the Stokes equation whose domain is discretized by two independent polygonal subdomains with different meshsizes. This causes a non-conformity at the intersection of the subdomains or leaves a gap (unmeshed region) between them. In order to appropriately couple these two different discretizations, we propose suitable transmission conditions to preserve the high order convergence of the scheme. Furthermore, stability estimates are established in order to show the well-posedness of the method and the error estimates. In particular, for smooth enough solutions, the L 2 norm of the errors associated to the approximations of the velocity gradient, the velocity and the pressure are of order h k + 1 , where h is the meshsize and k is the polynomial degree of the local approximation spaces. Moreover, the method presents superconvergence of the velocity trace and the divergence-free postprocessed velocity. Finally, we show numerical experiments that validate our theory and the capacities of the method. [ABSTRACT FROM AUTHOR]

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]

In this article, we introduce an error representation function to perform adaptivity in time of the recently developed time-marching Discontinuous Petrov–Galerkin (DPG) scheme. We first provide an analytical expression for the error that is the Riesz representation of the residual. Then, we approximate the error by enriching the test space in such a way that it contains the optimal test functions. The local error contributions can be efficiently computed by adding a few equations to the time-marching scheme. We analyze the quality of such approximation by constructing a Fortin operator and providing an a posteriori error estimate. The time-marching scheme proposed in this article provides an optimal solution along with a set of efficient and reliable local error contributions to perform adaptivity. We validate our method for both parabolic and hyperbolic problems. • We derive an error representation formula for adaptivity of the time-marching DPG scheme. • We provide the analytical error and an approximation of it by enriching the test space. • We introduce efficient and reliable local error contributions to perform adaptivity. • We analyze the approximation of the error by constructing a global Fortin operator. • We provide an a posteriori error estimate. • We validate our adaptive method for both parabolic and hyperbolic problems. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes a novel method to construct solid rational T-splines for complex genus-zero geometry from boundary surface triangulations. We first build a parametric mapping between the triangulation and the boundary of the parametric domain, a unit cube. After that we adaptively subdivide the cube using an octree subdivision, project the boundary nodes onto the input triangle mesh, and at the same time relocate the interior nodes via mesh smoothing. This process continues until the surface approximation error is less than a pre-defined threshold. T-mesh is then obtained by pillowing the subdivision result one layer on the boundary and its quality is improved. Templates are implemented to handle extraordinary nodes and partial extraordinary nodes in order to get a gap-free T-mesh. The obtained solid T-spline is -continuous except for the local region around each extraordinary node and partial extraordinary node. The boundary surface of the solid T-spline is -continuous everywhere except for the local region around the eight nodes corresponding to the eight corners of the parametric cube. Finally, a Bézier extraction technique is used to facilitate T-spline based isogeometric analysis. The obtained Bézier mesh is analysis-suitable with no negative Jacobians. Several examples are presented in this paper to show the robustness of the algorithm. [Copyright &y& Elsevier]

An effective discrete element modelling strategy for triangular mesh represented spherical harmonic particles is proposed. It features: (1) using a golden spiral lattice on the unit sphere to generate an initial triangular mesh with any number of vertices/triangles for a star-shaped surface; (2) applying an edge contraction mesh simplification algorithm to reduce the mesh size to any desired level; and (3) adopting an energy-conserving linear normal contact model to compute the contact geometric and force features of contacting particles. In particular, the edge contraction algorithm is applicable to any triangular mesh. It is algorithmically very simple and highly effective, and can be easily incorporated into existing discrete element frameworks. Numerical experiments are conducted to demonstrate that the simplified mesh by the edge contraction can not only have a very low geometric approximation error but also achieve expected mechanical responses. Thus this mesh simplification approach can serve as an ideal pre-processing tool to optimise a large input triangular mesh in order to significantly reduce the computational cost associated with discrete element simulations without compromising the modelling accuracy. • Established an effective triangular mesh representation procedure for star-shaped particles. • Introduced a highly effective edge-contraction based mesh simplification approach. • Described an energy-conserving contact model for triangulated particles. • Conducted numerical simulations to validate the proposed methodology. [ABSTRACT FROM AUTHOR]

Nondestructive damage identification is a central task, for example, in aeronautical, civil and naval engineering. The identification approaches based on (physical) models rely on the predictive accuracy of the forward model, and typically suffer from effects caused by ubiquitous modeling errors and uncertainties. The present paper considers the identification of defects in beams and plates under uncertain mass density distribution and present some examples using synthetic data. We show that conventional maximum likelihood and conventional maximum a posteriori approaches can yield unfeasible estimates in the presence of such uncertainties even when the actual damage can be parameterized/described only with a few parameters. To partially compensate for mass density uncertainties, we adopt the Bayesian approximation error approach (BAE) for inverse problems which is based on (approximative) marginalization over the model uncertainties. • Modeling uncertainties cause negative impacts on damage identification. • We propose an approach for damage identification under uncertain mass density fields. • Bayesian approximation error is used for compensating uncertainties. • Standard maximum likelihood provides unfeasible estimates under uncertainties. • Approximate marginalization over uncertainties provides feasible estimates. [ABSTRACT FROM AUTHOR]

The semiclassical Schrödinger equation with time-dependent potentials is an important model to study electron dynamics under external controls in the mean-field picture. In this paper, we propose two multiscale finite element methods to solve this problem. In the offline stage, for the first approach, the localized multiscale basis functions are constructed using sparse compression of the Hamiltonian operator at the initial time; for the latter, basis functions are further enriched using a greedy algorithm for the sparse compression of the Hamiltonian operator at later times. In the online stage, the Schrödinger equation is approximated by these localized multiscale basis functions in space and is solved by the Crank–Nicolson method in time. These multiscale basis functions have compact supports in space, leading to the sparsity of the stiffness matrix, and thus the computational complexity of these two methods is comparable to that of the standard finite element method. The spatial mesh size in the multiscale finite element methods is O (ε) , while the mesh size in the standard finite element method is o (ε) , where ε is the semiclassical parameter. Through a number of numerical examples in 1D and 2D, for approximately the same number of basis, we show that the approximation error of the multiscale finite element method is at least two orders of magnitude smaller than that of the standard finite element method, and the enrichment further reduces the error by another one order of magnitude. • Two multiscale methods are proposed to solve semiclassical Schrödinger equations with multiscale and time-dependent potentials. • A greedy algorithm to enrich multiscale basis functions and reduce the approximation error. • The computational complexity of our methods is comparable with the standard FEM. • More than two orders of magnitude reduction of the approximation error is achieved in our methods compared to the standard FEM. [ABSTRACT FROM AUTHOR]

Over the last few decades, Reduced Order Modeling (ROM) has slowly but surely inched towards widespread acceptance in computational mechanics, as well as other simulation-based fields. Projection-based Reduced Order Modeling (PROM) relies on the construction of an appropriate Reduced Basis (RB), which is typically a low-rank representation of a set of "observations" made using full-field simulations, usually obtained through truncated Singular Value Decomposition (SVD). However, SVD encounters limitations when dealing with a large number of high-dimensional observations, requiring the development of alternatives such as the incremental SVD. The key advantages of this approach are reduced computational complexity and memory requirement compared to a regular "single pass" spectral decomposition. These are achieved by only using relevant observations to enrich the low-rank representation as and when available, to avoid having to store them. In addition, the RB may be truncated 'on-the-fly' so as to reduce the size of the matrices involved as much as possible and, by doing so, avoid the quadratic scale-up in computational effort with the number of observations. In this paper, we present a new error estimator for the incremental SVD, which is shown to be an upper bound for the approximation error, and propose an algorithm to perform the incremental SVD truncation and observation selection 'on-the-fly', instead of using a prohibitively large number of frequently "hard to set" parameters. The performance of the approach is discussed on the reduced-order Finite Element (FE) model simulation of impact on a Taylor beam. • Incremental SVD implemented both in the offline and online phases of Galerkin PROM. • Simplified error estimator for the incremental SVD with 'On the fly' truncation. • Two problem-dependent parameters replaced by a single one. • Implementation in the legacy code RADIOSS. • Scalability and performance compared with Fortran Intel MKL. [ABSTRACT FROM AUTHOR]

We propose a conservative physics-informed neural network (cPINN) on discrete domains for nonlinear conservation laws. Here, the term discrete domain represents the discrete sub-domains obtained after division of the computational domain, where PINN is applied and the conservation property of cPINN is obtained by enforcing the flux continuity in the strong form along the sub-domain interfaces. In case of hyperbolic conservation laws, the convective flux contributes at the interfaces, whereas in case of viscous conservation laws, both convective and diffusive fluxes contribute. Apart from the flux continuity condition, an average solution (given by two different neural networks) is also enforced at the common interface between two sub-domains. One can also employ a deep neural network in the domain, where the solution may have complex structure, whereas a shallow neural network can be used in the sub-domains with relatively simple and smooth solutions. Another advantage of the proposed method is the additional freedom it gives in terms of the choice of optimization algorithm and the various training parameters like residual points, activation function, width and depth of the network etc. Various forms of errors involved in cPINN such as optimization, generalization and approximation errors and their sources are discussed briefly. In cPINN, locally adaptive activation functions are used, hence training the model faster compared to its fixed counterparts. Both, forward and inverse problems are solved using the proposed method. Various test cases ranging from scalar nonlinear conservation laws like Burgers, Korteweg–de Vries (KdV) equations to systems of conservation laws, like compressible Euler equations are solved. The lid-driven cavity test case governed by incompressible Navier–Stokes equation is also solved and the results are compared against a benchmark solution. The proposed method enjoys the property of domain decomposition with separate neural networks in each sub-domain, and it efficiently lends itself to parallelized computation, where each sub-domain can be assigned to a different computational node. • Domain decomposition technique is proposed for PINNs with tailored neural network in each sub-domain for solving conservation laws. • Due to presence of multiple neural networks, the representation capacity of the proposed cPINN method increases. • Based on prior knowledge of the solution regularity in each sub-domain, the hyper-parameter set of corresponding PINN can be properly adjusted. • The partial independence of individual PINNs in decomposed domains can be further employed to implement cPINN in a parallelized algorithm. [ABSTRACT FROM AUTHOR]

In this paper we propose a novel staggered discontinuous Galerkin method for the Brinkman problem on general polygonal meshes. The proposed method is robust in the Stokes and Darcy limits, in addition, hanging nodes can be automatically incorporated in the construction of the method, which are desirable features in practical applications. There are three unknowns involved in our formulation, namely velocity gradient, velocity and pressure. Unlike the original staggered DG formulation proposed for the Stokes equations in Kim et al. (2013), we relax the tangential continuity of velocity and enforce different staggered continuity properties for the three unknowns, which is tailored to yield an optimal L 2 error estimates for velocity gradient, velocity and pressure independent of the viscosity coefficient. Moreover, by choosing suitable projection, superconvergence can be proved for L 2 error of velocity. Finally, several numerical results illustrating the good performances of the proposed method and confirming the theoretical findings are presented. • The method is robust in both Stokes and Darcy limits, that is, the error of approximation is uniform with respect to the parameters in the equations. • A new idea to enforce continuity of numerical solutions across cell boundaries is developed. • The optimal convergence of the method is proved. Also, a superconvergence result is proved. • The method is very general, and it can be used with very general polygonal meshes. [ABSTRACT FROM AUTHOR]

We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that involves projecting the finite element solution onto a low-dimensional subspace and sketching the reduced equations using randomised sampling. We show that a sampling distribution based on the leverage scores of a tall matrix associated with the discrete Laplacian operator, can achieve nearly optimal performance and a significant speedup. We derive an expression of the complexity of the algorithm in terms of the number of samples that are necessary to meet an error tolerance specification with high probability, and an upper bound for the distance between the sketched and the high-dimensional solutions. Our analysis shows that the projection not only reduces the dimension of the problem but also regularises the reduced system against sketching error. Our numerical simulations suggest speed improvements of two orders of magnitude in exchange for a small loss in the accuracy of the prediction. • New randomised algebra approach to expedite the computation of the FEM. • Efficient sampling that preserves aspects of leverage score sampling. • Rigorous analysis of the errors incurred in the methodology. • For a moderate approximation error we obtain substantial time savings. [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]

The present paper investigates the use of different knot placement techniques for isogeometric analysis of spatial curved beams, to enhance analysis results in cases when geometries are given in terms of data points. Focusing on analysis-aware modeling for structural static and vibration simulations of spatial free-form curved beams, the knot placement techniques based on uniformly spaced knots as well as on De Boor's and Piegl and Tiller's algorithms are studied. For this purpose, an isogeometric formulation for linear Euler–Bernoulli beams based on the Euler–Rodriguez transformation rule is implemented. Different case studies and numerical examples are presented and the results are validated against "overkill" solutions computed with a commercial finite element software. The results show that the De Boor's knot placement algorithm typically leads to better approximation errors and is therefore the suggested strategy for this kind of problems. • Different knot placement techniques for isogeometric analysis of spatial curved beams are investigated. • This study is essential when the geometry needs to be constructed from input dataset by means of curve approximation. • Different knot placements not only affect the quality of the constructed geometry but also change the characteristics of the IGA elements and accuracy of results. • An isogeometric formulation for Euler–Bernoulli beams based on Euler–Rodriguez transformation is implemented. • The examples revealed that De Boor's algorithm typically leads to superior results compared to uniformly spaced and Piegl and Tiller's knot placement strategies. [ABSTRACT FROM AUTHOR]

Topology optimization considering material strength, structural stiffness and stability has great potential in engineering applications. In this paper, a topology optimization model considering the three vital performance indices is formulated by taking stress and buckling constrained continuum structures for compliance minimization. An effective optimization algorithm is developed for dealing with various issues and difficulties involved in the solution. The Kreisselmeier–Steinhauser (K–S) aggregation function is introduced to approximate the maximum von Mises stress together with a stability transformation method (STM) based correction scheme to reduce the approximation error. A smooth buckling aggregation function is constructed with a number of low-order buckling load factors to replace the original possibly non-smooth buckling constraints. To improve the robustness of the algorithm and achieve better designs, a continuation strategy for several optimization parameters is employed. With all these techniques, stable convergence of the iterative solution process is achieved. Three numerical examples are presented to demonstrate the effectiveness of the proposed algorithm. [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]