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The optimal Petrov–Galerkin formulation to solve partial differential equations (PDEs) recovers the best approximation in a specified finite-dimensional (trial) space with respect to a suitable norm. However, the recovery of this optimal solution is contingent on being able to construct the optimal weighting functions associated with the trial basis. While explicit constructions are available for simple one- and two-dimensional problems, such constructions for a general multidimensional problem remain elusive. In the present work, we revisit the optimal Petrov–Galerkin formulation through the lens of deep learning. We propose an operator network framework called Petrov–Galerkin Variationally Mimetic Operator Network (PG-VarMiON), which emulates the optimal Petrov–Galerkin weak form of the underlying PDE. The PG-VarMiON is trained in a supervised manner using a labeled dataset comprising the PDE data and the corresponding PDE solution, with the training loss depending on the choice of the optimal norm. The special architecture of the PG-VarMiON allows it to implicitly learn the optimal weighting functions, thus endowing the proposed operator network with the ability to generalize well beyond the training set. We derive approximation error estimates for PG-VarMiON, highlighting the contributions of various error sources, particularly the error in learning the true weighting functions. Several numerical results are presented for the advection-diffusion equation to demonstrate the efficacy of the proposed method. By embedding the Petrov–Galerkin structure into the network architecture, PG-VarMiON exhibits greater robustness and improved generalization compared to other popular deep operator frameworks, particularly when the training data is limited.

We develop a method for offline parameter estimation of time-discrete multibody dynamics in maximal coordinates with regularized and frictional kinematic constraints. This setting leads to unobserved degrees of freedom, which we handle using joint state and parameter estimation. Our method finds the states and parameters as the solution to a nonlinear least squares optimization problem based on the inverse dynamics and the observation error. The solution is found using a Levenberg–Marquardt algorithm with derivatives from automatic differentiation and custom differentiation rules for the complementary conditions that appear due to dry frictional constraints. We reduce the number of method parameters to the choice of the time-step, regularization coefficients, and a parameter that controls the relative weighting of inverse dynamics and observation errors. We evaluate the method using synthetic and real measured data, focusing on performance and sensitivity to method parameters. In particular, we optimize over a 13-dimensional parameter space, including inertial, frictional, tilt, and motor parameters, using data from a real Furuta pendulum. Results show fast convergence, in the order of seconds, and good agreement for different time-series of recorded data over multiple method parameter choices. However, very stiff constraints may cause difficulties in solving the optimization problem. We conclude that our method can be very fast and has method parameters that are robust and easy to set in the tested scenarios.

In this note, we introduce a novel stabilization mechanism for Cut Finite Element Methods (CutFEM), generalizing previous ghost penalty techniques in two key aspects: (1) the choice of stabilized quantities and (2) the selection of elements involved in stabilization. This approach notably allows for flexible and precise definitions of the stabilized quantities, including various functionals associated with the discrete solution, such as finite element degrees of freedom. We demonstrate that the kernel of our proposed ghost penalty operator defines a finite element space characterized by discrete extensions, closely related to those previously presented in Burman et al. (Numer Math 152(2):331–369, 2022).

We develop an interpolation-based modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using finite element methods, while the dependence on a finite-dimensional parameter is approximated separately. We establish existence, uniqueness, and regularity of the parametric solution and derive rigorous error estimates that explicitly quantify the interplay between spatial discretization and parameter approximation.In low-dimensional parameter spaces, classical interpolation schemes yield algebraic convergence rates based on Sobolev regularity in the parameter variable. In higher-dimensional parameter spaces, we replace classical interpolation by extreme learning machine (ELM) surrogates and obtain error bounds under explicit approximation and stability assumptions. The proposed framework is applied to inverse problems in quantitative photoacoustic tomography, where we derive potential and parameter reconstruction error estimates and demonstrate substantial computational savings compared to standard approaches, without sacrificing accuracy.

Cut finite element methods (CutFEM) extend the standard finite element method to unfitted meshes, enabling the accurate resolution of domain boundaries and interfaces without requiring the mesh to conform to them. This approach preserves the key properties and accuracy of the standard method while addressing challenges posed by complex geometries and moving interfaces.

In recent years, CutFEM has gained significant attention for its ability to discretize partial differential equations in domains with intricate geometries. This paper provides a comprehensive review of the core concepts and key developments in CutFEM, beginning with its formulation for common model problems and the presentation of fundamental analytical results, including error estimates and condition number estimates for the resulting algebraic systems. Stabilization techniques for cut elements, which ensure numerical robustness, are also explored. Finally, extensions to methods involving Lagrange multipliers and applications to time-dependent problems are discussed.

This paper addresses the problem of friction-free contact between two elastic bodies. We develop an augmented Lagrangian method that provides computational convenience by reformulating the contact problem as a nonlinear variational equality. To achieve this, we propose a Nitsche-based method incorporating a hybrid displacement variable defined on an interstitial layer. This approach enables complete decoupling of the contact domains, with interaction occurring exclusively through the interstitial layer. The layer is independently approximated, eliminating the need to handle intersections between unrelated meshes. Additionally, the method supports introducing an independent model on the interface, which we leverage to represent a membrane covering one of the bodies as well as a plate resting on one of the bodies. We present the formulation of the method, establish stability and error estimates, and demonstrate its practical utility through illustrative numerical examples.

We consider elliptic problems in multipatch isogeometric analysis (IGA) where the patch parameterizations may be singular. Specifically, we address cases where certain dimensions of the parametric geometry diminish as the singularity is approached — for example, a curve collapsing into a point (in 2D), or a surface collapsing into a point or a curve (in 3D). To deal with this issue, we develop a robust weak formulation for the second-order Laplace equation that allows trimmed (cut) elements, enforces interface and Dirichlet conditions weakly, and does not depend on specially constructed approximation spaces. Our technique for dealing with the singular maps is based on the regularization of the Riemannian metric tensor, and we detail how to implement this robustly. We investigate the method's behavior when applied to a square-to-cusp parameterization that allows us to vary the singular behavior's aggressiveness in how quickly the measure tends to zero when the singularity is approached. We propose a scaling of the regularization parameter to obtain optimal order approximation. Our numerical experiments indicate that the method is robust also for quite aggressive singular parameterizations.

We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of boundary observations for typical solutions (collective data) and a bulk measurement of a specific realization. To leverage this collective data, we first compress the boundary data using proper orthogonal decomposition (POD) in a linear expansion. Next, we identify a possible nonlinear low-dimensional structure in the expansion coefficients using an autoencoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train an operator network to map the expansion coefficients representing the boundary data to the finite element (FE) solution of the PDE. Finally, we connect the autoencoder's decoder to the operator network which enables us to solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method (FEM) in the linear setting and establish an optimal error estimate in the H¹-norm. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.

In this note, we design a cut finite element method for a low order divergence-free element applied to a boundary value problem subject to Stokes' equations. For the imposition of Dirichlet boundary conditions, we consider either Nitsche's method or a stabilized Lagrange multiplier method. In both cases, the normal component of the velocity is constrained using a multiplier, different from the standard pressure approximation. The divergence of the approximate velocities is pointwise zero over the whole mesh domain, and we derive optimal error estimates for the velocity and pressures, where the error constant is independent of how the physical domain intersects the computational mesh, and of the regularity of the pressure multiplier imposing the divergence-free condition.

We prove error estimates for a finite element approximation of viscoelastic dynamics based on continuous Galerkin in space and time, both in energy norm and in L2 norm. The proof is based on an error representation formula using a discrete dual problem and a stability estimate involving the kinetic, elastic, and viscoelastic energies. To set up the dual error analysis and to prove the basic stability estimates, it is natural to formulate the problem as a first-order-in-time system involving evolution equations for the viscoelastic stress, the displacements, and the velocities. The equations for the viscoelastic stress can, however, be solved analytically in terms of the deviatoric strain velocity, and therefore, the viscoelastic stress can be eliminated from the system, resulting in a system for displacements and velocities.