Galerkin Methods for PDEs: From Deterministic to Probabilistic
(2025) In Bachelor’s Theses in Mathematical Sciences NUMK11 20251Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
- Abstract
- Deterministic Galerkin finite element methods deliver high-fidelity solutions of partial differential equations (PDEs) but do not fully capture the inevitable uncertainty in material parameters, boundary conditions and external forcing. This thesis presents a unified framework that spans classical Galerkin discretisation and its stochastic counterpart, enabling uncertainty simulation of diffusion-type problems governed by elliptic PDEs. We begin with a rigorous derivation of the weak form for second-order elliptic PDEs, prove existence and uniqueness of the deterministic solution via the Lax–Milgram theorem, and recover first-order error estimates in the H1-norm. To incorporate randomness, the diffusion coefficient and source terms are... (More)
- Deterministic Galerkin finite element methods deliver high-fidelity solutions of partial differential equations (PDEs) but do not fully capture the inevitable uncertainty in material parameters, boundary conditions and external forcing. This thesis presents a unified framework that spans classical Galerkin discretisation and its stochastic counterpart, enabling uncertainty simulation of diffusion-type problems governed by elliptic PDEs. We begin with a rigorous derivation of the weak form for second-order elliptic PDEs, prove existence and uniqueness of the deterministic solution via the Lax–Milgram theorem, and recover first-order error estimates in the H1-norm. To incorporate randomness, the diffusion coefficient and source terms are parametrized as functions of both spatial variables and a finite set of random variables, defined over a complete probability space. This parameter-dependent formulation allows the use of Bochner spaces, whose Hilbert-tensor product structure naturally supports the construction of stochastic function spaces. Based on this foundation, we derive a stochastic weak formulation for which Lax-Milgram theorem ensures the existence and uniqueness of a solution with finite mean-square energy. The stochastic Galerkin method is implemented by tensorizing spatial finite element bases with orthogonal polynomial chaos bases in the stochastic domain, resulting in a high-dimensional block system with sparse matrices that captures both spa- tial resolution and the parameter-dependency in a single solve. Leveraging Céa’s lemma in the tensor-product setting, we establish mean-square quasi-optimality of the Galerkin solution. Numerical experiments in both Neumann and Dirichlet boundary conditions verify spectral convergence with respect to the stochastic discretization degree and confirm the theoretical error bounds. The proposed method provides not only accurate mean predictions but also precise quantification of variances and higher-order uncertainties, making it suitable for robust design which provides probabilistic quantification of uncertainty. (Less)
- Popular Abstract
- Imagine you’re modeling how heat travels through materials or water flows underground. Typically, scientists use mathematical equations called “partial differential equations” (PDEs) to describe these physical processes. These equations precisely represent how phenomena like heat, fluid dynamics, and electrical fields behave.
To solve these equations practically, researchers use an approach known as the Galerkin method. Rather than requiring exact solutions everywhere, this method simplifies the problem into an approximate version, called the “weak form,” making it much easier and efficient to handle numerically.
However, real-world materials such as concrete or soil don’t always behave predictably. Properties like thermal... (More) - Imagine you’re modeling how heat travels through materials or water flows underground. Typically, scientists use mathematical equations called “partial differential equations” (PDEs) to describe these physical processes. These equations precisely represent how phenomena like heat, fluid dynamics, and electrical fields behave.
To solve these equations practically, researchers use an approach known as the Galerkin method. Rather than requiring exact solutions everywhere, this method simplifies the problem into an approximate version, called the “weak form,” making it much easier and efficient to handle numerically.
However, real-world materials such as concrete or soil don’t always behave predictably. Properties like thermal conductivity or permeability vary, introducing uncertainty into the models. Ignoring these uncertainties can lead to inaccurate or unsafe predictions.
To address this, our research incorporates randomness into the Galerkin method, using what’s known as the “stochastic Galerkin method”. Instead of providing a single outcome, this method gives us a range of possible outcomes by modeling uncertain parameters as random variables. We use advanced mathematical techniques called polynomial chaos expansions to systematically capture this variability.
This approach doesn’t just provide average predictions; it also quantifies uncertainty. Engineers, scientists, and decision-makers can thus better understand potential risks, improving safety, efficiency, and reliability in their designs and operations. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9208294
- author
- Mo, Hailan and Yang, Yiwen LU
- supervisor
- organization
- course
- NUMK11 20251
- year
- 2025
- type
- M2 - Bachelor Degree
- subject
- publication/series
- Bachelor’s Theses in Mathematical Sciences
- report number
- LUNFNA-4064-2025
- ISSN
- 1654-6229
- other publication id
- 2025:K18
- language
- English
- id
- 9208294
- date added to LUP
- 2025-08-06 15:48:03
- date last changed
- 2025-08-06 15:48:03
@misc{9208294,
abstract = {{Deterministic Galerkin finite element methods deliver high-fidelity solutions of partial differential equations (PDEs) but do not fully capture the inevitable uncertainty in material parameters, boundary conditions and external forcing. This thesis presents a unified framework that spans classical Galerkin discretisation and its stochastic counterpart, enabling uncertainty simulation of diffusion-type problems governed by elliptic PDEs. We begin with a rigorous derivation of the weak form for second-order elliptic PDEs, prove existence and uniqueness of the deterministic solution via the Lax–Milgram theorem, and recover first-order error estimates in the H1-norm. To incorporate randomness, the diffusion coefficient and source terms are parametrized as functions of both spatial variables and a finite set of random variables, defined over a complete probability space. This parameter-dependent formulation allows the use of Bochner spaces, whose Hilbert-tensor product structure naturally supports the construction of stochastic function spaces. Based on this foundation, we derive a stochastic weak formulation for which Lax-Milgram theorem ensures the existence and uniqueness of a solution with finite mean-square energy. The stochastic Galerkin method is implemented by tensorizing spatial finite element bases with orthogonal polynomial chaos bases in the stochastic domain, resulting in a high-dimensional block system with sparse matrices that captures both spa- tial resolution and the parameter-dependency in a single solve. Leveraging Céa’s lemma in the tensor-product setting, we establish mean-square quasi-optimality of the Galerkin solution. Numerical experiments in both Neumann and Dirichlet boundary conditions verify spectral convergence with respect to the stochastic discretization degree and confirm the theoretical error bounds. The proposed method provides not only accurate mean predictions but also precise quantification of variances and higher-order uncertainties, making it suitable for robust design which provides probabilistic quantification of uncertainty.}},
author = {{Mo, Hailan and Yang, Yiwen}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor’s Theses in Mathematical Sciences}},
title = {{Galerkin Methods for PDEs: From Deterministic to Probabilistic}},
year = {{2025}},
}