Mixed-precision explicit stabilized Runge-Kutta methods for single- and multi-scale differential equations

Mixed-precision algorithms combine low-and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge-Kutta-Chebyshev (RKC) methods, where high precision is used for accuracy, and low precision for stability. Generally speaking, RKC methods are low-order explicit schemes with a stability domain growing quadratically with the number of function evaluations. For this reason, most of the computational effort is spent on stability rather than accuracy purposes. In this paper, we show that a naive mixed-precision implementation of any Runge-Kutta scheme can harm the convergence order of the method and limit its accuracy, and we introduce a new class of mixed precision RKC schemes that are instead unaffected by this limiting behavior. We present three mixed-precision schemes: a first-and a second-order RKC method, and a first order multirate RKC scheme for multiscale problems. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first-and second-order methods respectively) in high precision, while the rest are performed in low precision. We prove that while these methods are essentially as cheap as their fully low-precision equivalent, they retain the stability and convergence order of their high-precision counterpart. Indeed, numerical experiments confirm that these schemes are as accurate as the corresponding high-precision method. (C) 2022 Elsevier Inc. All rights reserved.

Adaptive Finite Element Methods for Multiscale Partial Differential Equations

Achim Nonnenmacher

Engineers rely on efficient simulations that provide them with reliable data in order to make proper engineering design decisions. The purpose of this thesis is to design adaptive numerical methods for multiscale problems in this spirit. We consider elliptic homogenization problems discretized by the finite element heterogeneous multiscale method (FE-HMM). Unlike standard (single-scale) finite element methods, our multiscale discretization scheme relies on coupled macro and micro finite elements. The framework of the HMM allows to design an algorithm that follows the classical finite element structure on the macro level. The fine scales of the multiscale problems are taken into account by replacing the element-wise numerical integration over unknown macroscopic data by a numerical integration over suitably averaged micro solutions. These micro solutions are obtained from micro FE problems on sampling domains within the macro elements. This thesis is divided into two parts. In the first part, we discuss a short and versatile FE implementation of the multiscale algorithm. The implementation is flexible, easy to use and to modify and can handle simplicial or quadrilateral FE and various macro-micro coupling conditions for the constrained micro problems. The implementation of time-dependent problems is also discussed. Numerical examples including three dimensional problems are presented and demonstrate the efficiency and the versatility of the computational strategy. In the second part (the main part of this thesis), we present an a posteriori error analysis for the FE-HMM. The a posteriori analysis enables us to estimate the accuracy of a numerical solution (and therefore its reliability) and further it allows for the design of adaptive numerical methods, which are the most efficient. The crucial component for the design of an adaptive multiscale method is the introduction of appropriate error indicators. As the error indicators depend on macroscopic data (such as the macroscopic diffusion tensor) that are not readily available, we construct error indicators that only depend on the available macro and micro FE solutions, available from previous computations. We provide a posteriori estimates for the upper and lower bound in the energy norm. The corresponding macroscopic mesh refinement strategy is therefore both reliable and efficient. The microscopic mesh is refined simultaneously and – under appropriate assumptions – optimally with the macroscopic mesh. This means that the strategy reduces the macro and micro error at the same rate. In the case of a uniformly oscillating tensor and exact micro computations, the standard a posteriori error estimates for the FEM applied to the homogenized problem are recovered. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method and demonstrate the optimality of the chosen macro-micro coupling. We extend the adaptive FE-HMM to higher order FE. We further derive a posteriori estimates for the error in quantities of interest that are needed to make certain design decisions; the quantity of interest is represented by a linear functional. We derive and analyze a multiscale counterpart to the classical dual-weighted residual method and design a corresponding goal-oriented adaptive multiscale method. The efficiency of the method is shown in numerical experiments.

EPFL2011

Novel corrector problems with exponential decay of the resonance error for numerical homogenization

Edoardo Paganoni

Multiscale problems, such as modelling flows through porous media or predicting the mechanical properties of composite materials, are of great interest in many scientific areas. Analytical models describing these phenomena are rarely available, and one must recur to numerical simulations. This represents a great computational challenge, because of the prohibitive computational cost of resolving the small scales. Multiscale numerical methods are therefore necessary to solve multiscale problems within reasonable computational time and resources. In particular, numerical homogenization techniques aim to capture the macroscopic behaviour with equations whose coefficients are computed numerically from the solutions of corrector problems at the microscale. A lack of knowledge of the coupling conditions between the micro- and the macro-scales brings in the so-called resonance error, which affects the accuracy of all multiscale methods. This source of error often dominates the numerical discretization errors and increasing its rate of decay is crucial for improving the accuracy of multiscale methods. In this work, we propose two novel upscaling schemes with arbitrarily high convergence rates of the resonance error to approximate the homogenized coefficients of scalar, linear second order elliptic differential equations. The first one is based on a parabolic equation, inspired by a model employed to compute the effective diffusive coefficients in stochastic diffusion processes. By using the approximation properties of smooth filtering functions, the homogenized coefficients can be approximated with arbitrary rates of accuracy. This claim is proved through an a priori convergence analysis, under the assumption of smooth periodic multiscale coefficients. Numerical experiments verify the expected convergence rates also under more general assumptions, such as discontinuous and random coefficients. The second method originates from the first by integrating the parabolic equation over a finite time interval. This method is referred to as the modified elliptic approach, because of the presence of a right-hand side which can be interpreted in terms of continuous semigroups and can be approximated numerically by Krylov subspace methods. The same convergence results as in the parabolic approach hold true. As a last step, a convergence analysis of the resonance error for the modified elliptic approach in the context of equations with random coefficients is performed. In this case, the resonance error is composed of a variance and a bias term, which can be bounded from above by a function decaying to zero. Numerical experiments reveal that the convergence rate of the resonance error for random coefficients is hampered, in comparison to the case of periodic coefficients, but the modified elliptic approach nevertheless outperforms standard methods.

EPFL2020

Numerical Methods for First and Second Order Fully Nonlinear Partial Differential Equations

Dimitrios Gourzoulidis

This thesis focuses on the numerical analysis of partial differential equations (PDEs) with an emphasis on first and second-order fully nonlinear PDEs. The main goal is the design of numerical methods to solve a variety of equations such as orthogonal maps, the prescribed Jacobian equation and inequality, the elliptic and parabolic Monge-Ampère equations. For orthogonal map we develop an \emph{operator-splitting/finite element} approach for the numerical solution of the Dirichlet problem. This approach is built on the variational principle, the introduction of an associated flow problem, and a time-stepping splitting algorithm. Moreover, we propose an extension of this method with an \emph{anisotropic mesh adaptation algorithm}. This extension allows us to track singularities of the solution's gradient more accurately. Various numerical experiments demonstrate the accuracy and the robustness of the proposed method for both constant and adaptive mesh. For the prescribed Jacobian equation and the three-dimensional Monge-Ampère equation, we consider a \emph{least-squares/relaxation finite element method} for the numerical solution of the Dirichlet problems. We then introduce a relaxation algorithm that splits the least-square problem, which stems from a reformulation of the original equations, into local nonlinear and variational problems. We develop dedicated solvers for the algebraic problems based on Newton method and we solve the differential problems using mixed low-order finite element method. Overall the least squares approach exhibits appropriate convergence orders in

L^2(\Omega)

and

H^1(\Omega)

error norms for various numerical tests. We also design a \emph{second-order time integration method} for the approximation of a parabolic two-dimensional Monge-Ampère equation. The space discretization of this method is based on low-order finite elements, and the time discretization is achieved by the implicit Crank-Nicolson type scheme. We verify the efficiency of the proposed method on time-dependent and stationary problems. The results of numerical experiments show that the method achieves nearly optimal orders for the

L^2(\Omega)

and

H^1(\Omega)

error norms when smooth solutions are approximated. Finally, we present an adaptive mesh refinement algorithm for the elliptic Monge-Ampere equation based on the residual error estimate. The robustness of the proposed algorithm is verified using various test cases and two different solvers which are inspired by the two previous proposed numerical methods.

EPFL2021