Goal-oriented Adaptive Finite Element Methods Research Articles

An Efficient Goal-Oriented Adaptive Finite Element Method for Accurate Simulation of Complex Electromagnetic Radiation Problems

A goal-oriented adaptive frequency domain finite element method for solving electromagnetic radiation problems including complex structures is presented in this paper. Compared with the traditional adaptive finite element method, the goal-oriented method can flexibly control the refined regions according to the parameters of interest; therefore it has better convergence and has made significant progress in scattering problems and eigenvalue problems. To simulate complex antennas, this paper proposes an error indicator with high accuracy and low computational cost, and it uses the adjoint problem to weight element residuals without additional degrees of freedom. Moreover, high-quality mesh refinement algorithms adapted to this indicator are developed using a suitable point insertion strategy for multiscale structures. By simulating two practical antennas, comparisons with the traditional goal-oriented FEM and the well-developed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> -adaptive finite element method in commercial software demonstrate the accuracy and efficiency of the proposed method.

EMFEM: A parallel 3D modeling code for frequency-domain electromagnetic method using goal-oriented adaptive finite element method

We have developed an open-source parallel 3D forward modeling code called EMFEM for the frequency-domain electromagnetic method (FDEM). This code is based on the adaptive finite element method and uses the total-field approach to solve the time-harmonic Maxwell equations. The use of tetrahedral meshes enables modeling of problems with complex geometry and topography. We utilize an iterative solver and an optimal block-diagonal preconditioner and auxiliary Maxwell solver to accelerate the convergence rate. We also use goal-oriented error estimation to adaptively refine the mesh and improve the accuracy of the solution. In addition, we develop a parallel algorithm based on the mesh partitioning approach. The effectiveness of the adaptive mesh refinement is demonstrated through a 1D layered model for marine controlled-source electromagnetic (CSEM) problems. We further present two 3D models for CSEM and magnetotelluric (MT) problems to study the scalability of the parallel algorithm, and the accuracy of the solutions is cross-validated with other open-source codes. The code is implemented in C++ and follows the modular design principle, making it easy to extend and maintain. The code is capable of modeling large-scale 3D problems and utilizing high-performance computing platforms. Furthermore, as the code is freely available, it can serve as a cross-validation tool for other 3D modeling codes and can also be utilized to implement 3D inversion routines.

Analyzing the effect of dipping anisotropy on 3D marine controlled-source electromagnetic response based on a goal-oriented adaptive finite element method

This paper presents a goal-oriented adaptive edge-based finite element (FE) algorithm for modelling marine controlled-source electromagnetic (CSEM) responses in three-dimensional (3-D) dipping anisotropic conductive media. We adopt the secondary field formulation for quasi-static Maxwell's equations to avoid the source singularity. The algorithm is implemented on unstructured tetrahedral grids, which allow for the simulation of complex model geometries, including tilted electrical interfaces and bathymetry. We validate the adaptive finite element code against a one-dimensional (1-D) layered model with vertical anisotropy. Further, we compare the 3-D FE numerical solutions with those obtained by a 2-D FE code for a two-dimensional dipping anisotropic model. The modelling results from the new 3-D FE algorithm are in good agreement with both the analytical solutions derived from the 1-D code and the numerical results from the anisotropic 2-D FE algorithm. We then use the algorithm to simulate the marine CSEM responses over different anisotropic 3-D models, including an anisotropic reservoir, anisotropic surrounding sediments, and an anisotropic overburden, and performed a detailed analysis of the effect of VTI, HTI, and TTI anisotropy on 3-D marine CSEM response. The modelling results illustrate that marine CSEM responses are considerably influenced by various anisotropic parameters, but to different extents. The research on different types of anisotropy within different media can help us avoid the misinterpretation of marine CSEM data when processing and interpreting of the measured geophysical electromagnetic data.

Goal-oriented adaptive finite element methods with optimal computational complexity

We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.

Rate-optimal goal-oriented adaptive FEM for semilinear elliptic PDEs

We formulate and analyze a goal-oriented adaptive finite element method for a semilinear elliptic PDE and a linear goal functional. The discretization is based on finite elements of arbitrary (but fixed) polynomial degree and involves a linearized dual problem. The linearization is part of the proposed algorithm, which employs a marking strategy different to that of standard adaptive finite element methods. Moreover, unlike the well-known dual-weighted residual strategy, the analyzed error estimators are completely computable. We prove linear convergence and, for the first time in the context of goal-oriented adaptivity for nonlinear PDEs, optimal algebraic convergence rates. In particular, the analysis does not require a sufficiently fine initial mesh.

A posteriori error estimates of goal-oriented adaptive finite element methods for nonlinear reaction–diffusion problems

In this paper, we construct two goal-oriented a posteriori error estimates with finite element approximations in the enriched space for the reaction–diffusion equation with the nonlinear reaction term. The first error estimator is proved to be the rigorous global lower and upper bounds on the error in the quantity of interest, and the second one is computed simply and efficiently. We propose the corresponding goal-oriented adaptive finite element algorithms, and show the effectiveness and the similar performance of those methods with a series of numerical experiments.

Multilevel correction goal-oriented adaptive finite element method for semilinear elliptic equations

In this study, a multilevel correction-type goal-oriented adaptive finite element method is designed for semilinear elliptic equations. Concurrently, the corresponding convergence property is theoretically proved. In the novel goal-oriented adaptive finite element method, only a linearized primal equation and a linearized dual equation are required to be solved in each adaptive finite element space. To ensure convergence, the approximate solution of the primal equation was corrected by solving a small-scale semilinear elliptic equation after the central solving process in each adaptive finite element space. Since solving of the large-scale semilinear elliptic equations is avoided and the goal-oriented technique is absorbed, there has been a significant improvement in the solving efficiency for the goal functional of semilinear elliptic equations.

Adaptive modelling of variably saturated seepage problems

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.

3-D direct current resistivity anisotropic modelling by goal-oriented adaptive finite element methods

3-D direct current resistivity anisotropic modelling by goal-oriented adaptive finite element methods

MARE2DEM: a 2-D inversion code for controlled-source electromagnetic and magnetotelluric data

This work presents MARE2DEM, a freely available code for 2-D anisotropic inversion of magnetotelluric (MT) data and frequency-domain controlled-source electromagnetic (CSEM) data from onshore and offshore surveys. MARE2DEM parametrizes the inverse model using a grid of arbitrarily shaped polygons, where unstructured triangular or quadrilateral grids are typically used due to their ease of construction. Unstructured grids provide significantly more geometric flexibility and parameter efficiency than the structured rectangular grids commonly used by most other inversion codes. Transmitter and receiver components located on topographic slopes can be tilted parallel to the boundary so that the simulated electromagnetic fields accurately reproduce the real survey geometry. The forward solution is implemented with a goal-oriented adaptive finite-element method that automatically generates and refines unstructured triangular element grids that conform to the inversion parameter grid, ensuring accurate responses as the model conductivity changes. This dual-grid approach is significantly more efficient than the conventional use of a single grid for both the forward and inverse meshes since the more detailed finite-element meshes required for accurate responses do not increase the memory requirements of the inverse problem. Forward solutions are computed in parallel with a highly efficient scaling by partitioning the data into smaller independent modeling tasks consisting of subsets of the input frequencies, transmitters and receivers. Non-linear inversion is carried out with a new Occam inversion approach that requires fewer forward calls. Dense matrix operations are optimized for memory and parallel scalability using the ScaLAPACK parallel library. Free parameters can be bounded using a new non-linear transformation that leaves the transformed parameters nearly the same as the original parameters within the bounds, thereby reducing non-linear smoothing effects. Data balancing normalization weights for the joint inversion of two or more data sets encourages the inversion to fit each data type equally well. A synthetic joint inversion of marine CSEM and MT data illustrates the algorithm's performance and parallel scaling on up to 480 processing cores. CSEM inversion of data from the Middle America Trench offshore Nicaragua demonstrates a real world application. The source code and MATLAB interface tools are freely available at http://mare2dem.ucsd.edu.

NCP Function--Based Dual Weighted Residual Error Estimators for Signorini's Problem

In this paper, we consider goal-oriented adaptive finite element methods for Signorini's problem. The basis is a mixed formulation, which is reformulated as nonlinear variational equality using a nonlinear complementarity function. For a general discretization, we derive error identities w.r.t. a possible nonlinear quantity of interest in the displacement as well as in the contact forces, which are included as Lagrange multiplier, using the dual weighted residual method. Afterwards, a numerical approximation of the error identities is introduced. We exemplify the results for a low order mixed discretization of Signorini's problem. The theorectical findings and the numerical approximation scheme are finally substantiated by some numerical examples.

A goal-oriented adaptive finite element method for 3D resistivity modeling using dual-error weighting approach

A goal-oriented adaptive finite element (FE) method for solving 3D direct current (DC) resistivity modeling problem is presented. The model domain is subdivided into unstructured tetrahedral elements that allow for efficient local mesh refinement and flexible description of complex models. The elements that affect the solution at each receiver location are adaptively refined according to a goal-oriented posteriori error estimator using dual-error weighting approach. The FE method with adapting mesh can easily handle such structures at almost any level of complexity. The method is demonstrated on two synthetic resistivity models with analytical solutions and available results from integral equation method, so the errors can be quantified. The applicability of the numerical method is illustrated on a resistivity model with a topographic ridge. Numerical examples show that this method is flexible and accurate for geometrically complex situations.

Goal-oriented adaptive finite element methods for elliptic problems revisited

A goal-oriented a posteriori error estimation of an output functional for elliptic problems is presented. Continuous finite element approximations are used in quadrilateral and triangular meshes. The algorithm is similar to the classical dual-weighted error estimation, however the dual weight contains solutions of the proposed patch problems. The patch problems are introduced to apply Clément and Scott–Zhang type interpolation operators to estimate point values with the finite element polynomials. The algorithm is shown to be reliable, efficient and convergent.

Convergence of goal-oriented adaptive finite element methods for semilinear problems

In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM (GOAFEM). Following the recent approach of Mommer-Stevenson and Holst-Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are then carefully examined. It is observed that the behavior of the implementation follows the predictions of the theory.

A hybrid boundary element-finite element approach to modeling plane wave 3D electromagnetic induction responses in the Earth

A novel hybrid boundary element-finite element scheme which is accelerated by an adaptive multi-level fast multipole algorithm is presented to simulate 3D plane wave electromagnetic induction responses in the Earth. The remarkable advantages of this novel scheme are the complete removal of the volume discretization of the air space and the capability of simulating large-scale complicated geo-electromagnetic induction problems. To achieve this goal, first the Galerkin edge-based finite-element method (FEM) using unstructured meshes is adopted to solve the electric field differential equation in the heterogeneous Earth, where arbitrary distributions of conductivity, magnetic permeability and dielectric permittivity are allowed for. Second, the point collocation boundary-element method (BEM) is used to solve a surface integral formula in terms of the reduced electrical vector potential on the arbitrarily shaped air–Earth interface. Third, to avoid explicit storage of the system matrix arising from large-scale problems and to reduce the horrendous time complexity of the product of the system matrix with an initial vector of unknowns, the adaptive multilevel fast multipole method is applied. This leads to a matrix-free form suitable for the application of iterative solvers. Furthermore, a highly sparse problem-dependent preconditioner is developed to significantly reduce the number of iterations used by the iterative solvers.The efficacy of the presented hybrid scheme is verified on two synthetic examples against different numerical techniques such as goal-oriented adaptive finite-element methods. Numerical experiments show that at low frequencies, where the quasi-static approximation is applicable, standard FEM methods prove to be superior to our hybrid BEM–FEM solutions in terms of computational time, because the FEM method requires only a coarse discretization of the air domain and offers an advantageous sparsity of the system matrix. At radio-magnetotelluric frequencies of a few hundred kHz, the hybrid BEM–FEM scheme outperforms the FEM method, because it avoids explicit storage of the system matrices as well as dense volume discretization of the air domain required by FEM methods at high frequencies. In summary, to the best of our knowledge, this study is the first attempt at completely removing the air space for large scale complicated electromagnetic induction modeling in the Earth.

A parallel goal-oriented adaptive finite element method for 2.5-D electromagnetic modelling

SUMMARY We present a parallel goal-oriented adaptive finite element method that can be used to rapidly compute highly accurate solutions for 2.5-D controlled-source electromagnetic (CSEM) and 2-D magnetotelluric (MT) modelling problems. We employ unstructured triangular grids that permit efficient discretization of complex modelling domains such as those containing topography, dipping layers and multiple scale structures. Iterative mesh refinement is guided by a goal-oriented error estimator that considers the relative error in the strike aligned fields and their spatial gradients, resulting in a more efficient mesh refinement than possible with a previous approach based on the absolute errors. Reliable error estimation is accomplished by a dual weighted residual method that is carried out via hierarchical basis computations. Our algorithm is parallelized over frequencies, wavenumbers, transmitters and receivers, where adaptive refinement is performed in parallel on subsets of these parameters. Mesh sharing allows an adapted mesh generated for a particular frequency and wavenumber to be shared with nearby frequencies and wavenumbers, thereby efficiently reducing the parallel load of the adaptive refinement calculations. We demonstrate the performance of our algorithm on a large cluster computer through scaling tests for a complex model that includes strong seafloor topography variations and multiple thin stacked hydrocarbon reservoirs. In tests using up to 800 processors and a realistic suite of CSEM data parameters, our algorithm obtained run-times as short as a few seconds to tens of seconds.

Weighted Marking for Goal-oriented Adaptive Finite Element Methods

We propose a new goal-oriented adaptive finite element method, which is based on the weighting of the residuals from the primal and dual problem at each step of the iteration. Our main result is the quasi-optimality of the resulting algorithm. Numerical experiments are reported, showing the convergence behavior of the algorithm.

A Goal-Oriented Adaptive Finite Element Method with Convergence Rates

An adaptive finite element method is analyzed for approximating functionals of the solution of symmetric elliptic second order boundary value problems. We show that the method converges and derive a favorable upper bound for its convergence rate and computational complexity. We illustrate our theoretical findings with numerical results.

Goal-oriented a posteriori error estimates in linear elastic fracture mechanics

Goal-oriented a posteriori error estimators are presented in this paper for the error obtained while approximately evaluating the J-integral in linear elastic fracture mechanics using the finite element method. The J-integral provides a criterion for crack propagation of a pre-existing crack. Therefore, its reliable and accurate evaluation is of considerable importance in engineering applications to prevent failure of structures. This can be achieved by goal-oriented adaptive finite element methods based on the well-established strategy of solving an auxiliary dual problem in order to control the error of a (non)linear functional, here the J-integral. In this paper, two principally different goal-oriented adaptive strategies are presented. The first strategy rests upon equilibrated residual error estimators based on local Neumann problems, whereas the second one makes use of averaging techniques. The paper is concluded by a numerical example that illustrates our theoretical results.