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This article summarizes the L p L^{p} -DPG method presented in [18], where only 1D convection-diffusion problems are solved. We apply the same computational techniques to 2D convection-diffusion problems and report additional numerical results herein. Furthermore, we propose an L p L^{p} -DPG method with variable 𝑝 and illustrate it with numerical experiments.

In this article, we present a general methodology to combine the Discontinuous Petrov-Galerkin (DPG) method in space and time in the context of methods of lines for transient advection-reaction problems. We rst introduce a semidiscretization in space with a DPG method rede ning the ideas of optimal testing and practicality of the method in this context. Then, we apply the recently developed DPG-based time-marching scheme, which is of exponential-type, to the resulting system of Ordinary Differential Equations (ODEs). We also discuss how to e ciently compute the action of the exponential of the matrix coming from the space semidiscretization without assembling the full matrix. Finally, we verify the proposed method for 1D+time advection-reaction problems showing optimal convergence rates for smooth solutions and more stable results for linear conservation laws comparing to the classical exponential integrators.

Following Muga and van der Zee (Muga and van der Zee, 2015), we generalize the standard Discontinuous Petrov–Galerkin (DPG) method, based on Hilbert spaces, to Banach spaces. Numerical experiments using model 1D convection-dominated diffusion problem are performed and compared with Hilbert setting. It is shown that Banach-based method gives solutions less susceptible to Gibbs phenomenon. h-adaptivity is implemented with the help of the error representation function as error indicator.

High-frequency wave propagation has many important applications in acoustics, elastodynamics, and electromagnetics. Unfortunately, the finite element discretization for these problems suffers from significant numerical pollution errors that increase with the wavenumber. It is critical to control these errors to obtain a stable and accurate method. We study the effect of pollution for very long waveguide problems in the context of robust discontinuous Petrov–Galerkin (DPG) finite element discretizations. Our numerical experiments show that the pollution primarily has a diffusive effect causing energy loss in the DPG method while phase errors appear less significant. We report results for 3D vectorial time-harmonic Maxwell problems in waveguides with more than 8000 wavelengths. Our results corroborate previous analysis for the Galerkin discretization of the Helmholtz operator by Melenk and Sauter (2011). Additionally, we discuss adaptive refinement strategies for multi-mode fiber waveguides where the propagating transverse modes must be resolved sufficiently. Our study shows the applicability of the DPG error indicator to this class of problems. Finally, we illustrate the importance of load balancing in these simulations for distributed-memory parallel computing.

We present an efficient implementation of the double adaptivity algorithm of Cohen et al. (2012) within the setting of the Petrov–Galerkin method with optimal test functions. We apply this method to the ultraweak variational formulation of a general linear variational problem discretized with the standard Galerkin finite element method. As an example, we demonstrate the feasibility of the method in the context of the convection-dominated diffusion problem. The presented ideas, however, apply to virtually any well-posed system of first-order partial differential equations, including singular perturbation problems.

The work is concerned with two problems: (a) analysis of a discontinuous Petrov–Galerkin (DPG) method set up in fractional energy spaces, (b) use of the results to investigate a non-conforming version of the DPG method for general polyhedral meshes. We use the ultraweak variational formulation for the model Laplace equation. The theoretical estimates are supported with 3D numerical experiments.

We propose an adaptive multigrid preconditioning technology for solving linear systems arising from Discontinuous Petrov–Galerkin (DPG) discretizations. Unlike standard multigrid techniques, this preconditioner involves only trace spaces defined on the mesh skeleton, and it is suitable for adaptive hp-meshes. The key point of the construction is the integration of the iterative solver with a fully automatic and reliable mesh refinement process provided by the DPG technology. The efficacy of the solution technique is showcased with numerous examples of linear acoustics and electromagnetic simulations, including simulations in the high-frequency regime, problems which otherwise would be intractable. Finally, we analyze the one-level preconditioner (smoother) for uniform meshes and we demonstrate that theoretical estimates of the condition number of the preconditioned linear system can be derived based on well established theory for self-adjoint positive definite operators.

We present both modeling and computational advancements to a unique three-dimensional discontinuous Petrov–Galerkin finite element model for the simulation of laser amplification in a fiber amplifier. Our model is based on the time-harmonic Maxwell equations, and it incorporates both amplification via an active dopant and thermal effects via coupling with the heat equation. As a full vectorial finite element simulation, this model distinguishes itself from other fiber amplifier models that are typically posed as an initial value problem and make significantly more approximations. Our model supports co-, counter-, and bi-directional pumping configurations, as well as inhomogeneous and anisotropic material properties. The long-term goal of this modeling effort is to study nonlinear phenomena that prohibit achieving unprecedented power levels in fiber amplifiers, along with validating typical approximations used in lower-fidelity models. The high-fidelity simulation comes at the cost of a high-order finite element discretization with many degrees of freedom per wavelength. This is necessary to counter the effect of numerical pollution due to the high-frequency nature of the wave simulation. To make the computation more feasible, we have developed a novel longitudinal model rescaling, using artificial material parameters with the goal of preserving certain quantities of interest. Our numerical tests demonstrate the applicability and utility of this scaled model in the simulation of an ytterbium-doped, step-index fiber amplifier that experiences laser amplification and heating. We present numerical results for the nonlinear coupled Maxwell/heat model with up to 240 wavelengths.

We construct a general family of DPG Fortin operators for the exact energy spaces defined on a tetrahedral element.

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 propose and investigate the application of alternative enriched test spaces in the discontinuous Petrov–Galerkin (DPG) finite element framework for singular perturbation linear problems, with an emphasis on 2D convection-dominated diffusion. Providing robust L 2 {L^{2}} error estimates for the field variables is considered a convenient feature for this class of problems, since this norm would not account for the large gradients present in boundary layers. With this requirement in mind, Demkowicz and others have previously formulated special test norms, which through DPG deliver the desired L 2 {L^{2}} convergence. However, robustness has only been verified through numerical experiments for tailored test norms which are problem-specific, whereas the quasi-optimal test norm (not problem specific) has failed such tests due to the difficulty to resolve the optimal test functions sought in the DPG technology. To address this issue (i.e. improve optimal test functions resolution for the quasi-optimal test norm), we propose to discretize the local test spaces with functions that depend on the perturbation parameter ϵ. Explicitly, we work with B-spline spaces defined on an ϵ-dependent Shishkin submesh. Two examples are run using adaptive h-refinement to compare the performance of proposed test spaces with that of standard test spaces. We also include a modified norm and a continuation strategy aiming to improve time performance and briefly experiment with these ideas.

Numerical integration of the stiffness matrix in higher-order finite element (FE) methods is recognized as one of the heaviest computational tasks in an FE solver. The problem becomes even more relevant when computing the Gram matrix in the algorithm of the Discontinuous Petrov Galerkin (DPG) FE methodology. Making use of 3D tensor-product shape functions, and the concept of sum factorization, known from standard high-order FE and spectral methods, here we take advantage of this idea for the entire exact sequence of FE spaces defined on the hexahedron. The key piece to the presented algorithms is the exact sequence for the one-dimensional element, and use of hierarchical shape functions. Consistent with existing results, the presented algorithms for the integration of H 1 {H^{1}} , H ⁢ ( curl ) {H(\operatorname{curl})} , H ⁢ ( div ) {H(\operatorname{div})} , and L 2 {L^{2}} inner products, have the 𝒪 ⁢ ( p 7 ) {\mathcal{O}(p^{7})} computational complexity in contrast to the 𝒪 ⁢ ( p 9 ) {\mathcal{O}(p^{9})} cost of conventional integration routines. Use of Legendre polynomials for shape functions is critical in this implementation. Three boundary value problems under different variational formulations, requiring combinations of H 1 {H^{1}} , H ⁢ ( div ) {H(\operatorname{div})} and H ⁢ ( curl ) {H(\operatorname{curl})} test shape functions, were chosen to experimentally assess the computation time for constructing DPG element matrices, showing good correspondence with the expected rates.

This article lays a mathematical foundation for goal-oriented adaptive mesh refinement and a posteriori error estimation with discontinuous Petrov--Galerkin (DPG) finite element methods. A goal-ori...

The finite element method (FEM) is a popular tool for solving engineering problems governed by Partial Differential Equations (PDEs). The accuracy of the numerical solution depends on the quality of the computational mesh. We consider the self-adaptive hp-FEM, which generates optimal mesh refinements and delivers exponential convergence of the numerical error with respect to the mesh size. Thus, it enables solving difficult engineering problems with the highest possible numerical accuracy. We replace the computationally expensive kernel of the refinement algorithm with a deep neural network in this work. The network learns how to optimally refine the elements and modify the orders of the polynomials. In this way, the deterministic algorithm is replaced by a neural network that selects similar quality refinements in a fraction of the time needed by the original algorithm.

Logarithmic conformation reformulations for viscoelastic constitutive laws have alleviated the high Weissenberg number problem, and the exploration of highly elastic flows became possible. However, stabilized formulations for logarithmic conformation reformulations in the context of finite element methods have not yet been widely studied. We present stabilized formulations for the logarithmic reformulation of the Oldroyd-B model by Saramito (2014) based on the Variational Multiscale framework and Galerkin/Least-Squares method. The reformulation allows the use of Newton's method due to its fully-implicit nature for solving the steady-state problem directly. The proposed stabilization methods cure instabilities of convection and compatibility while preserving a three-field problem. The spatial accuracy of the formulations is assessed with the four-roll periodic box, revealing comparable accuracy between the methods. The formulations are validated with benchmark flows past a cylinder and through a 4:1 contraction. We found an excellent agreement to benchmark results in the literature. The algebraic sub-grid scale method is highly robust, indicated by the high limiting Weissenberg numbers in the benchmark flows., Comment: submitted to JNNFM, 29 pages, 16 figures

We present a Petrov-Gelerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al.~(2012), we embed the original nonlocal convection-dominated diffusion problem into a larger mixed problem so as to choose an enriched test space as a stabilization of the numerical algorithm. In the numerical experiments, we use an approximate optimal test norm which can be efficiently implemented in 1d, and study its performance against the energy norm on the test space. We conduct convergence studies for the nonlocal problem using uniform $h$- and $p$-refinements, and adaptive $h$-refinements on both smooth manufactured solutions and solutions with sharp gradient in a transition layer. In addition, we confirm that the PG method is asymptotically compatible.

Least-squares (LS) and discontinuous Petrov–Galerkin (DPG) finite element methods are an emerging methodology in the computational partial differential equations with unconditional stability and built-in a posteriori error control. This special issue represents the state of the art in minimal residual methods in the L 2 L^{2} -norm for the LS schemes and in dual norm with broken test-functions in the DPG schemes.

The Discontinuous Petrov–Galerkin (DPG) method for high frequency wave propagation problems is discussed. The DPG method, with its attractive uniform (mesh and wavenumber independent) pre-asymptotic stability property, allows for a fully automatic adaptive h p -algorithm that can be initiated from very coarse meshes. Moreover, DPG always delivers a Hermitian positive definite system, suggesting the use of the Conjugate Gradient algorithm for its solution. We present a new iterative solution scheme which capitalizes on these attractive properties of DPG. This novel solver is integrated within the adaptive procedure by constructing a two-grid-like preconditioner for the Conjugate Gradient method that exploits information from previous meshes. The construction of our preconditioner is discussed, and its efficacy is illustrated with an example of a 2D acoustics problem. Our results show that the proposed iterative algorithm converges at a rate independent of the mesh and the wavenumber.

A discontinuous Petrov-Galerkin (DPG) method is used to solve the time-harmonic equations of linear viscoelasticity. It is based on a "broken" primal variational formulation, which is very similar to the classical primal variational formulation used in Galerkin methods, but has additional "interface" variables at the boundaries of the mesh elements. Both the classical and broken formulations are proved to be well-posed in the infinite-dimensional setting, and the resulting discretization is proved to be stable. A full $hp$-convergence analysis is also included, and the analysis is verified using computational simulations. The method is particularly useful as it carries its own natural arbitrary-$p$ a posteriori error estimator, which is fundamental for solving problems with localized solution features. This proves to be useful when validating calibration models of dynamic mechanical analysis (DMA) experiments. Indeed, different DMA experiments of epoxy and silicone resins were successfully validated to within $5\%$ of the quantity of interest using the numerical method., 23 pages, 5 figures

The use of “ideal” optimal test functions in a Petrov–Galerkin scheme guarantees the discrete stability of the variational problem. However, in practice, the computation of the ideal optimal test functions is computationally intractable. In this paper, we study the effect of using approximate, “practical” test functions on the stability of the DPG (discontinuous Petrov–Galerkin) method and the change in stability between the “ideal” and “practical” cases is analyzed by constructing a Fortin operator. We highlight the construction of an “optimal” DPG Fortin operator for H 1 and H ( div ) spaces; the continuity constant of the Fortin operator is a measure of the loss of stability between the ideal and practical DPG methods. We take a two-pronged approach: first, we develop a numerical procedure to estimate an upper bound on the continuity constant of the Fortin operator in terms of the inf–sup constant γ h of an auxiliary problem. Second, we construct a sequence of approximate Fortin operators and exactly compute the continuity constants of the approximate operators, which provide a lower bound on the exact Fortin continuity constant. Our results shed light not only on the change in stability by using practical test functions, but also indicate how stability varies with the approximation order p and the enrichment order Δ p . The latter has important ramifications when one wishes to pursue local h p -adaptivity.

We explore a vexing benchmark problem for viscoelastic fluid flows with the discontinuous Petrov-Galerkin (DPG) finite element method of Demkowicz and Gopalakrishnan [1,2]. In our analysis, we develop an intrinsic a posteriori error indicator which we use for adaptive mesh generation. The DPG method is useful for the problem we consider because the method is inherently stable---requiring no stabilization of the linearized discretization in order to handle the advective terms in the model. Because stabilization is a pressing issue in these models, this happens to become a very useful property of the method which simplifies our analysis. This built-in stability at all length scales and the a posteriori error indicator additionally allows for the generation of parameter-specific meshes starting from a common coarse initial mesh. A DPG discretization always produces a symmetric positive definite stiffness matrix. This feature allows us to use the most efficient direct solvers for all of our computations. We use the Camellia finite element software package [3,4] for all of our analysis., 20 pages, 18 figures, 6 tables

We propose a three dimensional Discontinuous Petrov-Galerkin Maxwell approach for modeling Raman gain in fiber laser amplifiers. In contrast with popular beam propagation models, we are interested in a truly full vectorial approach. We apply the ultraweak DPG formulation, which is known to carry desirable properties for high-frequency wave propagation problems, to the coupled Maxwell signal/pump system and use a nonlinear iterative scheme to account for the Raman gain. This paper also introduces a novel and practical full-vectorial formulation of the electric polarization term for Raman gain that emphasizes the fact that the computer modeler is only given a measured bulk Raman gain coefficient. Our results provide promising qualitative corroboration of the model and methodology used., Comment: 32 pages, 15 figures, submitted to Journal of Computational Physics

Discontinuous Petrov Galerkin (DPG) methods are made easily implementable using `broken' test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact sequence of first order (unbroken) Sobolev spaces are of particular interest. A characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided. Stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces. This technique is used to provide a complete error analysis of DPG methods for Maxwell equations with perfect electric boundary conditions. The technique also permits considerable simplifications of previous analyses of DPG methods for other equations. Reliability and efficiency estimates for an error indicator also follow. Finally, the equivalence of stability for various formulations of the same Maxwell problem is proved, including the strong form, the ultraweak form, and a spectrum of forms in between.

Higher order finite element (FE) methods provide significant advantages in a number of applications such as wave propagation, where high order shape functions help to mitigate pollution (dispersion) error. However, classical assembly of higher order systems is computationally burdensome, requiring the evaluation of many point quadrature schemes. When the Discontinuous Petrov-Galerkin (DPG) FE methodology is employed, the use of an enriched test space further increases the computational burden of system assembly, increasing the relevance of improved assembly techniques. Sum factorization—a technique that exploits the tensor-product structure of shape functions to accelerate numerical integration—was proposed in Ref. [10] for the assembly of DPG matrices on hexahedral elements that reduced the computational complexity from order O ( p 9 ) to O ( p 7 ) (where p denotes polynomial order). In this work we extend the concept of sum factorization to the construction of DPG matrices on prismatic elements by expressing prism shape functions as tensor products of 2D simplex and 1D interval shape functions. Unexpectedly, the resulting sum factorization routines on partially-tensorized prism shape functions achieve the same O ( p 7 ) complexity as sum factorization on fully-tensorized hexahedra shape functions (as products of 1D interval shape functions) presented in Ref. [10]. Throughout this work we adhere to the theory of exact sequence energy spaces, proposing sum factorization routines for each of the 3D FE exact sequence energy spaces—H1, H(curl), H(div), and L2. Computational results for construction of the DPG Gram matrix on a prismatic element in each exact sequence energy space are presented, corroborating the expected O ( p 7 ) complexity. Additionally, construction of the DPG system for an ultraweak Maxwell problem on a prismatic element is considered and a partially-tensorized sum factorization for hexahedral elements is proposed to improve implementational compatibility between hexahedral and prismatic elements.

In this article, several discontinuous Petrov-Galerkin (DPG) methods with perfectly matched layers (PMLs) are derived along with their quasi-optimal graph test norms. Ultimately, two different complex coordinate stretching strategies are considered in these derivations. Unlike with classical formulations used by Bubnov-Galerkin methods, with so-called ultraweak variational formulations, these two strategies in fact deliver different formulations in the PML region. One of the strategies, which is argued to be more physically natural, is employed for numerically solving two- and three-dimensional time-harmonic acoustic, elastic, and electromagnetic wave propagation problems, defined in unbounded domains. Through these numerical experiments, efficacy of the new DPG methods with PMLs is verified., 19 pages, 6 figures

This article introduces the DPG-star (from now on, denoted DPG$^*$) finite element method. It is a method that is in some sense dual to the discontinuous Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG$^*$ methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG$^*$ and DPG methods can be seen as generalizations of $\mathcal{L}\mathcal{L}^\ast$ and least-squares methods, respectively. A priori error analysis and a posteriori error control for the DPG$^*$ method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG$^*$ and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable.

In this paper we propose a new algorithm for solving of challenging adaptive time-dependent problems with Crank–Nicolson kind of time integration in parallel. The new algorithm allows for parallel execution of computations from different time steps. Time steps are distributed between processors. The number of processors working over consecutive time steps increases with each iteration of the adaptive algorithm. The following time steps utilize the previous time steps's solutions with the same level of accuracy. Our new parallel algorithm is compared with other methods. First, we compare it with a traditional method which performs all the adaptive iterations in the first time step, next it restarts the adaptive iterations in the second time step, and continues, one time step after another. Second, we compare our algorithm with the one that performs all the adaptive iterations in the first time step, and then starts the following time steps with the optimal mesh obtained from the previous iteration. Finally, we compare our algorithm to the one that executes the projection-based interpolation of the material data in the first time step, then it solves the problem over the obtained mesh, and then starts the following time steps with the optimal mesh obtained from the previous iteration. All the mentioned algorithms are tested on the challenging computational problem, which is the solution of the Pennes equation over a human head. The heat source is obtained by approximation of the solution of the Maxwell equation computed over the model human head. From our numerical results it follows that 10 min (600 s) of exposure to the cell phone radiation may cause up to 2 °C increase of the temperature of the brain in the range close to the cell phone.

The discontinuous Petrov-Galerkin methodology with optimal test functions (DPG) of Demkowicz and Gopalakrishnan 18,20 guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. Whereas Bubnov-Galerkin methods use identical trial and test spaces, Petrov-Galerkin methods allow these function spaces to differ. In DPG, test functions are computed on the fly and are chosen to realize the supremum in the inf-sup condition; the method is equivalent to a minimum residual method. For well-posed problems with sufficiently regular solutions, DPG can be shown to converge at optimal rates-the inf-sup constants governing the convergence are mesh-independent, and of the same order as those governing the continuous problem 48. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements.We employ DPG to solve the steady incompressible Navier-Stokes equations in two dimensions, building on previous work on the Stokes equations, and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh. We apply our approach to a manufactured solution due to Kovasznay as well as the lid-driven cavity flow, backward-facing step, and flow past a cylinder problems.

A unified construction of high order shape functions is given for all four classical energy spaces ($H^1$, $H(\mathrm{curl})$, $H(\mathrm{div})$ and $L^2$) and for elements of "all" shapes (segment, quadrilateral, triangle, hexahedron, tetrahedron, triangular prism and pyramid). The discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element. The shape functions are conforming, hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes. Expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape. The polynomial order is allowed to differ for each separate topological entity (vertex, edge, face or interior) in the mesh, so the shape functions can be used to implement local $p$ adaptive finite element methods. Each topological entity may have its own orientation, and the shape functions can have that orientation embedded by a simple permutation of arguments., Comment: 156 pages, 39 figures, 10 tables; corrected minor typos and misprints

This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually well-posed family of broken variational formulations of the original PDE. It can be exploited to solve challenging problems in a variety of physical scenarios where stability or a particular mode of convergence is desired in a part of the domain. The linear elasticity equations are solved in this work, but the approach can be applied to other equations as well. The broken variational formulations, which are essentially extensions of more standard formulations, are characterized by the presence of mesh-dependent broken test spaces and interface trial variables at the boundaries of the elements of the mesh. This allows necessary information to be naturally transmitted between adjacent subdomains, resulting in coupled variational formulations which are then proved to be globally well-posed. They are solved numerically using the DPG methodology, which is especially crafted to produce stable discretizations of broken formulations. Finally, expected convergence rates are verified in two different and illustrative examples., Comment: 24 pages, 5 figures

This work represents the first endeavor in using ultraweak formulations to implement high-order polygonal finite element methods via the discontinuous Petrov-Galerkin (DPG) methodology. Ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space, while most of the trial space can be chosen as copies of $L^2$-discretizations that have no need to be continuous across adjacent elements. Additionally, the test spaces are broken along the mesh interfaces. This allows one to construct conforming polygonal finite element methods, termed here as PolyDPG methods, by defining most spaces by restriction of a bounding triangle or box to the polygonal element. The only variables that require nontrivial compatibility across elements are the so-called interface or skeleton variables, which can be defined directly on the element boundaries. Unlike other high-order polygonal methods, PolyDPG methods do not require ad hoc stabilization terms thanks to the crafted stability of the DPG methodology. A proof of convergence of the form $h^p$ is provided and corroborated through several illustrative numerical examples. These include polygonal meshes with $n$-sided convex elements and with highly distorted concave elements, as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid. Since PolyDPG methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes. This work is also accompanied by an open-source $\texttt{PolyDPG}$ software supporting polygonal and conventional elements., 33 pages, 16 figures

A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz map on the test space. The resulting linear system is overdetermined. Two different approaches for solving the system are suggested (although others are discussed): (1) solving the associated normal equation with linear solvers for symmetric positive-definite systems (e.g. Cholesky factorization); and (2) solving the overdetermined system with orthogonalization algorithms (e.g. QR factorization). The finite element assembly algorithm for each of these approaches is described in detail. The normal equation approach is usually faster for direct solvers and requires less storage. The second approach reduces the condition number of the system by a power of two and is less sensitive to round-off error. The rectangular stiffness matrix of second approach is demonstrated to have condition number $\mathcal{O}(h^{-1})$ for a variety of formulations of Poisson's equation. The stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least-squares finite element methods and it is proved that the two are identical in some cases. An example with Poisson's equation indicates that the solutions of these two different linear systems can be nearly indistinguishable (if round-off error is not an issue) and rapidly converge to each other. The orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems. Experiments with Poisson's equation in single-precision arithmetic as well as the linear acoustics problem near resonance in double-precision arithmetic verify this conclusion., Comment: 30 pages

We develop a locally conservative formulation of the discontinuous Petrov-Galerkin finite element method (DPG) for convection-diffusion type problems using Lagrange multipliers to exactly enforce conservation over each element. We provide a proof of convergence as well as extensive numerical experiments showing that the method is indeed locally conservative. We also show that standard DPG, while not guaranteed to be conservative, is nearly conservative for many of the benchmarks considered. The new method preserves many of the attractive features of DPG, but turns the normally symmetric positive-definite DPG system into a saddle point problem.

The paper presents a frequency-domain h p -adaptive Finite Element (FE) Method for a class of coupled acoustics/elasticity/poroelasticity problems with application to modeling of acoustic logging measurements in complex borehole environments. The paper extends methodology, software, and results presented in Matuszyk et al. (2012). We derive for poroelastic media a mixed weak formulation, which is closely related to the enhanced ( u , p ) formulation developed by Atalla et al. (2001). The formulation is extended to the transversely anisotropic (VTI) case, and makes use of the viscodynamic operator, which enables more accurate calculations for higher frequencies. In this approach, a special split of the differential operator results in boundary and coupling conditions that can be easily implemented within the FE framework. The appropriate PML technique is employed to truncate the computational domain. Accordingly, the original h p -FE code is modified by the implementation of the appropriate scaling of the independent variables to make efficient the automatic adaptation algorithm. Solutions of non-trivial examples involving formations with permeable layers are presented. They positively verify the consistency and accuracy of the presented method.

The Discontinuous Petrov–Galerkin (DPG) method is a class of novel higher order adaptive finite element methods derived from the minimization of the residual of the variational problem (Demkowicz and Gopalakrishnan, 2011) [1], and has been shown to deliver a method for convection–diffusion that is provably robust in the diffusion parameter (Demkowicz and Heuer, in press; Chan et al., in press) [2,3]. In this work, the DPG method is extrapolated to nonlinear systems, and applied to several problems in fluid dynamics whose solutions exhibit boundary layers or singularities in stresses. In particular, the effectiveness of DPG as a numerical method for compressible flow is assessed with the application of DPG to two model problems over a range of Mach numbers and laminar Reynolds numbers using automatic adaptivity with higher order finite elements, beginning with highly under-resolved coarse initial meshes.

The subject of this article is a review of all possible transmission problems for electromagnetic phenomena. In particular, we study the case of a perfect dielectric and a perfect conductor via a (formal) limit with conductivity approaching zero or infinity and discuss the expected regularity of the involved unknowns. Finally, we formulate equivalent variational formulations for each considered problem.

We discuss well-posedness and convergence theory for the DPG method applied to a general system of linear Partial Differential Equations (PDEs) and specialize the results to the classical Stokes problem. The Stokes problem is an iconic troublemaker for standard Bubnov-Galerkin methods; if discretizations are not carefully designed, they may exhibit non-convergence or locking. By contrast, DPG does not require us to treat the Stokes problem in any special manner. We illustrate and confirm our theoretical convergence estimates with numerical experiments.

We introduce a DPG method for convection-dominated diffusion problems. The choice of a test norm is shown to be crucial to achieving robust behavior with respect to the diffusion parameter (Demkowicz and Heuer 2011) [18]. We propose a new inflow boundary condition which regularizes the adjoint problem, allowing the use of a stronger test norm. The robustness of the method is proven, and numerical experiments demonstrate the method's robust behavior.

A combination of ideas in least-squares finite element methods with those of hybridized methods recently led to discontinuous Petrov--Galerkin (DPG) finite element methods. They minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard, locally computable, dual norm. This paper establishes a general a posteriori error analysis for the natural norms of the DPG schemes under conditions equivalent to a priori stability estimates. It is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors. The presented abstract framework for a posteriori error analysis applies to known DPG discretizations of Laplace and Lame equations and to a novel DPG method for the stress-velocity formulation of Stokes flow with symmetric stress approximations. Since the error control does not rely on the discrete equations, it applies to inexactly computed or otherwise perturbed solutions within the discrete space...

We show that it is possible to apply the DPG methodology without reformulating a second-order boundary value problem into a first-order system, by considering the simple example of the Poisson equation. The result is a new weak formulation and a new DPG method for the Poisson equation, which has no numerical trace variable, but has a numerical flux approximation on the element interfaces, in addition to the primal interior variable.

We propose a unified discontinuous Petrov--Galerkin (DPG) framework with optimal test functions for Friedrichs-like systems, which embrace a large class of elliptic, parabolic, and hyperbolic partial differential equations (PDEs). The well-posedness, i.e., existence, uniqueness, and stability, of the DPG solution is established on a single abstract DPG formulation, and two abstract DPG methods corresponding to two different, but equivalent, norms are devised. We then apply the single DPG framework to several linear(ized) PDEs including, but not limited to, scalar transport, Laplace, diffusion, convection-diffusion, convection-diffusion-reaction, linear(ized) continuum mechanics (e.g., linear(ized) elasticity, a version of linearized Navier--Stokes equations, etc.), time-domain acoustics, and a version of the Maxwell's equations. The results show that we not only recover several existing DPG methods, but also discover new DPG methods for both PDEs currently considered in the DPG community and new ones. As ...

We propose and analyze a discontinuous Petrov--Galerkin (DPG) method for convection-dominated diffusion problems that provides robust $L^2$ error estimates for the field variables which are quasi-optimal in the energy norm. A key feature of the method is to construct test functions defined by a variational formulation with bilinear form (test norm) specifically designed for the goal of robustness. A main theoretical ingredient is a stability analysis of the adjoint problem. Numerical experiments underline our theoretical results and, in particular, confirm robustness of the DPG method for well-chosen test norms.