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Stokes flow equations have been implemented successfully in practice for simulating problems with moving interfaces. Though computational methods produce accurate solutions and numerical convergence can be demonstrated using a resolution…
Many surface reconstruction methods incorporate normal integration, which is a process to obtain a depth map from surface gradients. In this process, the input may represent a surface with discontinuities, e.g., due to self-occlusion. To…
We introduce a novel monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. This problem is prevalent in…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics-Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
Elasticity theory is an important component of continuum mechanics and has had widely spread applications in science and engineering. Material interfaces are ubiquity in nature and man-made devices, and often give rise to discontinuous…
Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as…
We propose an Extended Hybrid High-Order scheme for the Poisson problem with solution possessing weak singularities. Some general assumptions are stated on the nature of this singularity and the remaining part of the solution. The method is…
In this article, we introduce a new partially penalized immersed finite element method (IFEM) for solving elliptic interface problems with multi-domains and triple-junction points. We construct new IFE functions on elements intersected with…
In this work, we develop interface-gated physics-informed neural networks (IG-PINNs) to solve elliptic interface equations. In IG-PINNs, we use a fully connected neural network to capture the smooth behavior across the entire domain. In…
The solution of the elliptic partial differential equation has interface singularity at the points which are either the intersections of interfaces or the intersections of interfaces with the boundary of the domain. The singularities that…
Fluid-structure interaction models are used to study how a material interacts with different fluids at different Reynolds numbers. Examining the same model not only for different fluids but also for different solids allows to optimize the…
This paper studies a low-communication algorithm for solving elliptic partial differential equations (PDE's) on high-performance machines, the nested iteration with range decomposition algorithm (NIRD). Previous work has shown that NIRD…
The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet for fluid problems it only achieves first-order spatial accuracy near embedded boundaries for the velocity field and…
We propose a consistent physics-informed neural networks (CPINNs) framework for elliptic obstacle problems formulated as variational inequalities. The method is based on a mixed loss functional that is rigorously aligned with the stability…
We consider the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic problems can be decoupled via the introduction of a new unknown, corresponding to…
This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semilinear partial differential equation (PDE) with low-regular solutions in two/three dimensions. The method is obtained by combining…
Physics-informed neural networks (PINN) have achieved notable success in solving partial differential equations (PDE), yet solving the Navier-Stokes equations (NSE) with complex boundary conditions remains a challenging task. In this paper,…
Despite the extensive application of nonlinear Model Predictive Control (MPC) in automated driving, balancing its computational efficiency with respect to the control performance and constraint satisfaction remains a challenge in emergency…